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\title{M\'emoire d'habilitation}
\author{Stefan Neuwirth}
\date{}


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\begin{document}
\selectlanguage{frenchb}
\thispagestyle{empty}
\leavevmode\vskip 0pt plus 2fill
\begin{center}\parindent0pt

{\Huge M\'emoire}\vfill

d\'epos\'e en vue de l'obtention de\vfill

{\Huge l'habilitation \`a diriger des recherches}\vfill

par \vfill

Stefan Neuwirth\vskip 0pt plus 2fill

{\Huge Aspects quantitatifs de l'inconditionnalit\'e}

\end{center}\vskip 0pt plus 2fill

et soutenu le 9 d\'ecembre 2008 devant le jury compos\'e de
\begin{itemize}
\item Christian Le Merdy,
\item Daniel Li,
\item Gilles Pisier et
\item \'Eric Ricard,
\end{itemize}

les rapporteurs \'etant
\begin{itemize}
\item Aline Bonami,
\item Vern Paulsen et
\item Vladimir Peller.
\end{itemize}\vskip 0pt plus 4fill
\cleardoublepage

\pagestyle{headings}
\tableofcontents*\clearpage
\chapter*{Liste des articles et notes}\addcontentsline{toc}{chapter}{Liste des articles et notes}

{\renewcommand{\labelenumi}{[\Alph{enumi}]}
  \begin{enumerate}
  \item A.~Harcharras, S.~Neuwirth et K.~Oleszkiewicz, {\em
      \href{http://dx.doi.org/10.1512/iumj.2001.50.1952}{Lacunary
        matrices}}, Indiana Univ. Math. J. 50 (2001), 1675--1689.

  \item S.~Neuwirth, {\em
      \href{http://dx.doi.org/10.1017/S0024611506015899} {Cycles and
        1-unconditional matrices}}, Proc. London Math. Soc. (3) 93
    (2006), 761--790.

  \item W.~Banks, A.~Harcharras, S.~Neuwirth et {\'E}.~Ricard, {\em
      \href{http://dx.doi.org/10.1016/S0024-3795(02)00517-7}{Matrix
        inequalities with applications to the theory of iterated
        kernels}}, Linear Algebra Appl.  362 (2003), 275--286.

  \item {\sloppy S.~Neuwirth, {\em The size of bipartite graphs with girth
      eight}.  \href{http://arxiv.org/abs/math/0102210}
    {\texttt{http\string:/\kern-2pt/\allowbreak arxiv.\allowbreak org/\allowbreak
        abs/\allowbreak math/\allowbreak 0102210}}, 2001.

  }\item N.~Kosmatov et S.~Neuwirth, {\em Ordering simultaneously the
      columns and lines of 0,1~matrices}, manuscrit (2002).

  \item S.~Neuwirth et {\'E}.~Ricard, {\em Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group}, manuscrit (2008).

  \item S.~Neuwirth, {\em The {S}idon constant of sets with three
      elements}.
    \href{http://arxiv.org/abs/math/0102145}{\texttt{http\string:/\kern-2pt/\allowbreak arxiv.\allowbreak org/\allowbreak
        abs/\allowbreak math/\allowbreak0102145}}, 2001.

  \item \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em The
      maximum modulus of a trigonometric trinomial},
    J. Anal. Math. 104 (2008), 371--396.

  \item \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em On
      the Sidon constant of $\{0,1,2,3\}$}, manuscrit (2003).

  \item \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em
      \href{http://matwbn.icm.edu.pl/ksiazki/sm/sm131/sm13112.pdf}
      {Metric unconditionality and {F}ourier analysis}}, Studia
    Math. 131 (1998), 19--62.

  \item \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em
      \href{http://www.numdam.org/item?id=AIF_1999__49_6_1853_0}{Two
        random constructions inside lacunary sets}},
    Ann. Inst. Fourier (Grenoble) 49 (1999), 1853--1867.

  \end{enumerate}}\clearpage
%\nocite{A,B,C,D,E,F,G,H,I,Y,Z}\bibliography{neuwirth}
% \chapter{Th{\`e}se d'universit{\'e}}

% \section{R{\'e}sum{\'e} de la th{\'e}matique}

% Ma th\`ese propose l'\'etude de plusieurs propri\'et\'es
% fonctionnelles d'inconditionnalit\'e en les exprimant \`a l'aide de
% multiplicateurs.  

% La premi\`ere partie est consacr\'ee \`a l'\'etude
% de ph\'enom\`enes d'inconditionnalit\'e isom\'etrique et
% presqu'iso\-m\'etrique dans les espaces de Banach s\'eparables. Parmi
% ceux-ci, la notion la plus g\'en\'erale est celle de
% \emph{propri\'et\'e d'approximation inconditionnelle m\'etrique}. Elle
% se caract\'erise parmi les espaces de Banach de cotype fini par une
% propri\'et\'e simple d'<<~inconditionnalit\'e par blocs.~>> Cette
% propri\'et\'e est \'etudi\'ee pour les sous-espaces des espaces de
% Banach de fonctions sur le cercle qui sont engendr\'es par une suite
% de caract\`eres $\e^{\iu nt}$: dans ce cas, elle s'exprime en termes
% de multiplicateurs de Fourier. Les \emph{suites basiques inconditionnelles
% isom\'etriques} et \emph{presqu'isom\'etriques} de caract\`eres, en
% particulier les ensembles de Sidon de constante asymptotiquement 1, sont aussi \'etudi\'es.
% Chacun de ces cas est d\'ecrit par une propri\'et\'e combinatoire sur la
% suite de caract\`eres. 

% Une deuxi\`eme partie \'etudie le rapport entre la croissance d'une
% suite d'entiers et les propri\'et\'es harmoniques et fonctionnelles de
% la suite de caract\`eres associ\'ee. Nous montrons en particulier que
% toute suite polynomiale, ainsi que la suite des nombres premiers,
% contient un ensemble $E$ tel que
% \begin{itemize}
% \item toute fonction int\'egrable \`a spectre de Fourier dans $E$ est en fait
% $p$-int\'egrable pour tout $p<\infty$, mais
% \item il existe une fonction mesurable born\'ee \`a spectre de Fourier dans $E$
%   qui n'a pas de version continue.
% \end{itemize}
\chapter*[R\'esum\'e]{R\'esum\'e}
\addcontentsline{toc}{chapter}{R\'esum\'e}
\section{Trois cas de figure pour l'inconditionnalit\'e}
\label{sec::1}
\noindent Toutes nos recherches sont li\'ees \`a la notion d'inconditionnalit\'e,
motiv\'ee par la question suivante:
\begin{Q}
\label{q:inc}
  Lorsqu'un \'el\'ement~$x$ d'un espace norm\'e admet une
  repr\'esentation comme combinaison lin\'eaire $\sum c_q\e_q$
  d'\'el\'ements~$\e_q$, de quelle mani\`ere la norme de~$x$
  d\'epend-elle du signe des coefficients~$c_q$?
\end{Q}

Les r\'eponses que nous obtiendrons seront en termes du \emph{support}
de~$x$, c'est-\`a-dire de l'ensemble~$I$ d'indices~$q$ pour lesquels
$c_q\ne0$. 

 Selon la situation, un changement du signe
des coefficients~$c_q$
\begin{itemize}
\item fait varier la norme de~$x$ de mani\`ere born\'ee et on dira que
  $(\e_q)_{q\in I}$ est une \emph{suite basique inconditionnelle};
\item multiplie au plus la norme de~$x$ par un facteur~$D$ explicite
  et $D$~sera la \emph{constante d'inconditionnalit\'e} de la suite
  $(\e_q)_{q\in I}$;
\item ne change pas la norme de~$x$ et on parlera de suite basique $(\e_q)_{q\in I}$
  \emph{1-\hspace{0pt}inconditionnelle};
\end{itemize}

Lorsque nous chercherons \`a d\'eterminer une constante
d'inconditionnali\'e exacte, nous devrons pr\'eciser de quelle
mani\`ere nous nous permettons de changer le signe des coefficients:
\begin{itemize}\item 
  de mani\`ere \emph{r\'eelle} en multipliant certains coefficients
  par~$-1$, ou
\item de mani\`ere \emph{complexe} en les faisant tourner
  d'un angle~${t_q}$.
\end{itemize}
Voici
trois cas de figure dans lesquels cette question se pose.

\begin{enumerate}
\item Si $x$~est une fonction sur un groupe ab\'elien compact et les
  $\e_q$~sont les caract\`eres de ce groupe, cette repr\'esentation
  est la s\'erie de Fourier de~$x$ et un changement du signe des
  coefficients de Fourier est une \emph{convolution} ou \emph{multiplication de Fourier} unimodulaire.
\item Si $x$~est un op\'erateur et les $\e_q$~sont les matrices
  \'el\'ementaires, cette repr\'esentation est la matrice de~$x$ et un
  changement du signe des coefficients matriciels est une
  \emph{multiplication de Schur}  unimodulaire.
\item Si $x$~est un \'el\'ement de l'alg\`ebre d'un groupe discret~$G$
  et les $\e_q$~sont les fonctions indicatrices des \'el\'ements de~$G$, alors un changement du signe des coefficients est une
  \emph{multiplication de Herz-Schur} unimodulaire.
\end{enumerate}



\section{D\'efinition des suites basiques inconditionnelles}

Voici une d\'efinition formelle qui reprend la discussion ci-dessus.

\begin{dfnf}
  \label{1:unc:def}
  Soit~$X$ un espace vectoriel quasi-norm\'e muni d'une suite
  distingu\'ee~$(\e_q)$ dans~$X$. Soit~$(\e_q)_{q\in I}$ une
  sous-suite.  Soit~$\Sign=\T=\{z\in\C:\abs{z}=1\}$ (vs.~$\Sign=\{-1,1\}$.)
  \begin{enumerate}
  \item $I$~est \emph{inconditionnelle} dans~$X$ s'il y a une constante~$D$ telle que
    \begin{equation}\label{1:u}
      \Bigl\|\sum_{q\in I}\epsilon_qa_q\e_q\Bigr\| \le D
      \Bigl\|\sum_{q\in I} a_q\e_q\Bigr\|
    \end{equation}
    pour tout choix de signes~$\epsilon_q\in\Sign$ et toute suite de
    coefficients complexes~$a_q$ de support fini. Sa \emph{constante
      d'inconditionnalit\'e complexe \klammer{vs.~r\'eelle}} est le
    minimum de telles constantes~$D$.
\item $I$~est \emph{1-inconditionnelle} complexe
  \klammer{vs.~r\'eelle} dans~$X$ si sa constante
  d'inconditionnalit\'e complexe \klammer{vs.~r\'eelle} vaut~1%; on
%  d\'efinit de m\^eme les suites \emph{compl\`etement}
%  1-inconditionnelles complexes et r\'eelles. 
  . Cela veut
  dire que l'in\'egalit\'e~\eqref{1:u} devient l'\'egalit\'e
  \begin{equation*}
    \Bigl\|\sum_{q\in I}\epsilon_qa_q\e_q\Bigr\| =
    \Bigl\|\sum_{q\in I} a_q\e_q\Bigr\|.
  \end{equation*}
\end{enumerate}
\end{dfnf}

\section[Matrices et multiplicateurs de Schur pour les classes de Schatten]%
{Matrices et multiplicateurs de Schur pour les classes de Schatten-von-Neumann}

\noindent Cette th\'ematique de recherche correspond au cas de figure~$(b)$
ci-dessus: $(\e_q)$~est la suite des matrices \'el\'ementaires.

Notons~$\Col$ l'ensemble des indices colonne et~$\Row$ l'ensemble des
indices ligne de matrices, en g\'en\'eral deux copies de~$\N$, et soit~$I$ une partie de~$\Row\times \Col$. La propri\'et\'e
d'inconditionnalit\'e de~$I$ peut aussi se formuler ainsi:  une suite
$I$~est inconditionnelle de constante~$D$ si et seulement si, pour toute matrice~$\varphi$ \`a
coefficients complexes (vs.~r\'eels) et pour tout~$x$ dont les
coefficients de matrice sont nuls hors de~$I$ (dont l'espace sera
not\'e~$X_I$) on a
\begin{equation*}
  \|\varphi*x\|\le D\sup|\varphi_{rc}|\,\|x\|,
\end{equation*}
o\`u $\varphi*x$~est le produit de \emph{Schur} (ou de Hadamard) d\'efini par
\begin{equation*}
  (\varphi*x)_{rc}=\varphi_{rc}x_{rc}.
\end{equation*}

L'op\'erateur de multiplication par~$\ph$ est un \emph{multiplicateur
  de Schur relatif}. On peut aussi d\'ecrire les multiplicateurs de
Schur relatifs comme les op\'erateurs diagonaux sur la suite~$(\e_q)_{q\in I}$.

Notre \'etude se concentrera sur les classes de Schatten-von-Neumann
$X=\Sch^p$ dont la quasi-norme est donn\'ee par
$\|x\|={(\tr{(x^*x)}^{p/2})}^{1/p}$: il s'agit de la contrepartie
non commutative des espaces $\ell^p$. Lorsque $p\ge1$, l'espace de
Banach $\Sch^p$ admet une structure d'espace d'op\'erateurs
canonique qui rend la d\'efinition suivante naturelle.

\begin{dfnf}
  $I$~est \emph{compl\`etement inconditionnelle} dans~$\Sch^p$ s'il y
  a une constante~$D$ telle que \eqref{1:u} vaut pour tout choix de
  signes~$\epsilon_q\in\Sign$ et toute suite de coefficients
  \emph{op\'erateurs}~$a_q\in \Sch^p$ \`a support fini. Sa constante
  d'inconditionnalit\'e \emph{compl\`ete} complexe
  \klammer{vs.~r\'eelle} est le minimum de telles constantes~$D$.
\end{dfnf}
De la m\^eme mani\`ere, on parle de la norme \emph{compl\`ete} de
multiplicateurs de Schur relatifs.  On ne sait pas si on d\'efinit
vraiment ainsi une classe nouvelle; ce serait r\'epondre \`a la
conjecture de Gilles Pisier qu'il existe des multiplicateurs de Schur
born\'es sur~$S^p$ ($p\ne1,2,\infty$) qui ne sont pas compl\`etement born\'es.

Une suite inconditionnelle de matrices \'el\'ementaires dans~$\Sch^\infty$ est en fait un ensemble~$\Varopoulos$-Sidon, classe que
Varopoulos a introduite dans l'\'etude des alg\`ebres tensorielles~$\cnaught(\Col)\projtens \cnaught(\Row)$ sur des espaces discrets
(voir le th\'eor\`eme~\ref{cycl}.\ref{varo} page~\pageref{varo} qui
rassemble les r\'esultats connus: $I$~doit \^etre r\'eunion finie
d'ensembles qui soit contiennent au plus un \'el\'ement par ligne,
soit contiennent au plus un \'el\'ement par colonne.)  Notre \'etude
g\'en\'eralise ainsi les r\'esultats de Varopoulos \`a toutes les
classes de Schatten-von-Neumann.


\section{Suites de matrices \'el\'ementaires et graphes bipartis}

\noindent Les suites inconditionnelles de matrices \'el\'ementaires forment la
contrepartie matricielle des ensembles~$\Lambda(p)$ de Walter Rudin
\'etudi\'es en analyse de Fourier (le cas de figure~$(a)$ de la
section~\ref{sec::1}.) Alors que l'\'etude des ensembles~$\Lambda(p)$
voit surgir naturellement leurs propri\'et\'es arithm\'etiques (de
th\'eorie additive des nombres,) l'inconditionnalit\'e de~$I$ se
traduit avantageusement en termes de th\'eorie des graphes.


Nous allons donc consid\'erer~$I$ comme un graphe biparti dont les
deux classes (<<~couleurs~>>) de sommets sont~$\Col$ et~$\Row$, dont les \'el\'ements
seront appel\'es respectivement <<~sommets~colonne~>> et <<~sommets~ligne.~>> Ses
ar\^etes (non dirig\'ees) relient seulement des sommets~ligne $r\in \Row$ avec des
sommets~colonne $c\in \Col$, et cela exactement lorsque $(r,c)\in I$. 
La matrice~$(\chi_I(r,c))_{(r,c)\in\Row\times\Col}$ fonction
caract\'eristique de~$I$ est la matrice d'incidence de ce graphe
biparti.

Voici deux exemples importants.
\begin{exe}\label{2gra}
  Soit~$s$ un entier. Consid\'erons l'ensemble
    \begin{equation*}
      I=\bigl\{\,(r,c)\in\Z/s\Z\times\Z/s\Z:r-c\in\{0,1\}\,\bigr\}.
    \end{equation*}
    Le graphe biparti associ\'e est le cycle (ligne~0, colonne~0,
    ligne~1, colonne~1, \ldots, ligne~$s-1$, colonne~$s-1$) de
    longueur~$2s$.  La matrice d'incidence de ce graphe est
    \[
    \bordermatrix{
      &\scriptstyle0&\scriptstyle1&\scriptstyle\cdots&\scriptstyle
      s-2&\scriptstyle s-1\cr
      \hfil\scriptstyle0&1&0&\ddots&0&1\cr
      \hfil\scriptstyle1&1&1&\ddots&0&0\cr
      %\hfil\scriptstyle2&0&1&1&\ddots&0&0\cr
      \hfil\scriptstyle\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\cr
      \hfil\scriptstyle s-2&0&0&\ddots&1&0\cr \hfil\scriptstyle
      s-1&0&0&\ddots&1&1\cr}\text.
    \]
  \end{exe}
\begin{exe}\label{2grb}
  Consid\'erons l'ensemble
    \begin{equation*}
      I=\bigl\{\,(r,c)\in\Z/7\Z\times\Z/7\Z:r+c\in\{0,1,3\}\,\bigr\}.
    \end{equation*}
    Le graphe biparti associ\'e est le graphe de \emph{Heawood}.\vskip\abovedisplayskip
    \hfil\hbox{\includegraphics{heawood}}\vskip\belowdisplayskip
    La matrice d'incidence de ce graphe est
    \[
    \bordermatrix{
      &\scriptstyle0&\scriptstyle1&\scriptstyle2&\scriptstyle3&\scriptstyle4&\scriptstyle5&\scriptstyle6\cr
      \scriptstyle0&1&1&0&1&0&0&0\cr \scriptstyle1&1&0&1&0&0&0&1\cr
      \scriptstyle2&0&1&0&0&0&1&1\cr \scriptstyle3&1&0&0&0&1&1&0\cr
      \scriptstyle4&0&0&0&1&1&0&1\cr \scriptstyle5&0&0&1&1&0&1&0\cr
      \scriptstyle6&0&1&1&0&1&0&0\cr}.
    \]
 \end{exe}

\section{Matrices lacunaires et inconditionnalit\'e}

\noindent Dans l'article \emph{Lacunary matrices}, nous montrons que
ces sous-suites doivent satisfaire la condition de densit\'e suivante,
qui est l'analogue de la condition de maille de Walter Rudin
\cite[Theorem~3.5]{ru60} pour les ensembles~$\Lambda(p)$.
\begin{thmf}[page~\pageref{dc}]
  \label{densf}
  Si $I$~est inconditionnelle de constante~$D$ dans~$\Sch^p$, alors la
  taille~$\#I'$ de tout sous-graphe~$I'$ induit par $m$~sommets
  colonne et $n$~sommets ligne, c'est-\`a-dire le cardinal de toute
  partie $I'=I\cap \Row'\times \Col'$ avec~$\#\Col'=m$ et~$\#\Row'=n$,
  satisfait
  \begin{align}
    \#I'
    &\le D^2\bigl(m^{1/p}n^{1/2}+m^{1/2}n^{1/p}\bigr)^2\label{1:dens}\\
    \notag&\le4D^2\min(m,n)^{2/p}\max(m,n).
  \end{align}
  Les exposants de cette in\'egalit\'e sont optimaux dans les trois
  cas suivants:
  \begin{enumerate}
  \item si $m$~ou~$n$ est fix\'e (trivial;)
  \item si $p=4$ (voir le graphe al\'eatoire ci-dessous;)
  \item si $p$ est un entier pair et~$m=n$ (voir~\cite[Theorem~4.8]{ha98}.)
  \end{enumerate}
\end{thmf}

Si~$m\ne n$, nous construisons des graphes al\'eatoires qui testent
l'in\'egalit\'e \eqref{1:dens} sans en montrer toujours l'optimalit\'e.

\begin{thmf}[page~\pageref{randcon}]
  Pour tout~$\eps>0$ et tout entier pair~$p\ge4$, il existe un graphe~$I$ de taille
  \begin{equation*}
    \#I\sim
    \begin{cases}
      \max(m,n)^{1-\eps}\min(m,n)^{1/2}&\text{si $p=4$}\\
      \max(m,n)^{1/2-\eps}\min(m,n)^{1/2+2/p}&\text{si $p\ge6$.}
    \end{cases}
  \end{equation*}
  et de constante d'inconditionnalit\'e ind\'ependante de~$m$ et~$n$
  lorsque~$mn\to\infty$.
\end{thmf}
 
Si~$p$ est un entier pair, nous donnons aussi une condition suffisante
en termes de sentiers sur un graphe biparti: un \emph{sentier} de longueur~$s$ dans~$I$ est une suite~$(v_0,v_1,\dots,v_s)$ de sommets
alternativement dans~$\Row$ et~$\Col$ telle que les ar\^etes reliant
$v_0$ \`a~$v_1$, $v_1$ \`a~$v_2$, etc.\ correspondent \`a des
\'elements deux \`a deux distincts de~$I$ (alors qu'un \emph{chemin} est
requis d'avoir m\^eme tous ses sommets distincts et que les sommets
d'une \emph{promenade} sont admis \`a se r\'ep\'eter.) Le th\'eor\`eme
suivant est aussi l'analogue d'un th\'eor\`eme de Walter Rudin.

\begin{thmf}[page~\pageref{MAIN}]
  \label{sentf}
  Soit~$p$ un entier pair.  Si le nombre de sentiers dans~$I$ de
  longueur~$p/2$ entre deux sommets donn\'es admet une borne uniforme,
  alors $I$~est inconditionnelle dans~$\Sch^p$.
\end{thmf}

Le calcul suivant montre le lien \'etroit entre la norme~$S^p$
avec~$p=2s$ un entier pair et les promenades ferm\'ees de longueur~$p$
dans ce graphe.
\begin{align*}
  \tr\Bigl|\sum_{q\in I}a_q\e_q\Bigr|^p
  &=\tr\Bigl(\sum_{(r,c),(r',c')\in I} (a_{rc}\e_{rc})^*
  (a_{r'c'}\e_{r'c'})
  \Bigr)^s\\
  &=\tr~\sum_{\substack{(r_1,c_1),(r'_1,c'_1),\dots,\\
      (r_s,c_s),(r_s',c'_s)\in I}} ~\prod_{i=1}^s
  (\overline{a_{r_ic_i}}\e_{c_ir_i})
  (a_{r'_ic'_i}\e_{r'_ic'_i})\\
  &=\sum_{\substack{(r_1,c_1),(r_1,c_2),\dots,\\
      (r_s,c_s),(r_s,c_{s+1})\in I}} ~\prod_{i=1}^s
  \overline{a_{r_ic_i}}a_{r_ic_{i+1}}\quad(\text{o\`u}~c_{s+1}=c_1.)
\end{align*}
Cette derni\`ere somme est index\'ee par les  promenades ferm\'ees~$(c_1,r_1,\allowbreak c_2,\allowbreak \dots,\allowbreak c_s,r_s)$ de longueur~$p$ dans le graphe associ\'e
\`a~$I$!

La conjonction des th\'eor\`emes~\ref{densf} et~\ref{sentf} donne une
nouvelle preuve du th\'eor\`eme de Paul Erd\H{o}s selon lequel un
graphe sur $v$~sommets sans circuit de longueur paire~$p$ est de
taille born\'ee par~$v^{1+2/p}$, \`a une constante pr\`es (un \emph{circuit}
est un sentier ferm\'e.) La g\'en\'eralisation de ce th\'eor\`eme des
circuits aux \emph{cycles} (chemins ferm\'es) par Bondy et Simonovits
\cite{bs74} \'echappe \`a notre m\'ethode.  L'existence de graphes qui
montreraient l'optimalit\'e de cette estimation est une question
ouverte pos\'ee par Erd\H os en~1963.

\section[Matrices lacunaires et {\small1}-inconditionnalit\'e]{\texorpdfstring{Matrices lacunaires et 1-inconditionnalit\'e}{Matrices lacunaires et 1-inconditionnalit\'e}}

\noindent L'article \emph{Cycles and 1-unconditional matrices} aboutit
\`a une caract\'erisation des suites basiques
1-in\-con\-di\-tion\-nelles dans~$\Sch^p$.

Un des ingr\'edients est l'\'etude des multiplicateurs de Schur
unimodulaires sur un cycle. Nous obtenons en particulier la
proposition suivante.

\begin{prp}[page~\pageref{cyno}]
  Si~$p$ n'est pas un entier pair, alors $\epsilon$~est un
  multiplicateur de Schur unimodulaire isom\'etrique sur un cycle~$I$
  pour~$\Sch^p$ si et seulement si~$\epsilon$ peut \^etre interpol\'ee
  par une matrice de rang~1: $\epsilon_{rc}=\zeta_c\eta_r$ pour~$(r,c)\in I$, o\`u~$\zeta\in\T^\Col$ et~$\eta\in\T^\Row$.
\end{prp}

\begin{proof}[Esquisse de d\'emonstration] La condition est bien
  suffisante: on a alors
  \begin{equation*}
    \epsilon*x=
    \begin{pmatrix}
      \ddots&&\\&\eta_r&\\&&\ddots
    \end{pmatrix}
    (x_{rc})
    \begin{pmatrix}
      \ddots&&\\&\zeta_c&\\&&\ddots
    \end{pmatrix}.
  \end{equation*}
  \'Etudions la n\'ecessit\'e.  On peut supposer que le cycle~$I$ soit
  donn\'e comme dans l'exemple~\ref{2gra}. Soit~$\epsilon\in\T^I$ une
  matrice de nombres unimodulaires partiellement sp\'ecifi\'ee. Il est
  possible de multiplier les lignes et les colonnes de~$\epsilon$ par
  des nombres complexes de module~1 de sorte que $\epsilon$~devienne
  la matrice circulante
  \begin{equation}\label{multcyc}
    \tilde\epsilon=\bordermatrix{
      &\scriptstyle0&\scriptstyle1&\scriptstyle\cdots&\scriptstyle
      s-2&\scriptstyle s-1\cr 
      \hfil\scriptstyle0&1&0&\ddots&0&\th\cr
      \hfil\scriptstyle1&\th&1&\ddots&0&0\cr
      %\hfil\scriptstyle2&0&\th&\ddots&0&0\cr
      \hfil\scriptstyle\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\cr
      \hfil\scriptstyle s-2&0&0&\ddots&1&0\cr 
      \hfil\scriptstyle
      s-1&0&0&\ddots&\th&1\cr}
  \end{equation}
  avec $\th$~racine $s$i\`eme de~$\overline{\epsilon_{00}}
  \epsilon_{10} \dots \overline{\epsilon_{{s-1},{s-1}}}
  \epsilon_{0,{s-1}}$. Un argument de transfert montre que la norme du
  multiplicateur de Schur par~$\tilde\epsilon$ sur~$I$ borne le multiplicateur de
  Fourier relatif~$\mu\colon a+bz\mapsto a+\th bz$ dans le groupe~$G$
  des racines~$s$i\`emes de l'unit\'e, o\`u on norme~$a+bz$ par la
  norme~$\Ell^p$:
  $\|a+bz\|=\left(s^{-1}\sum_{z^s=1}|a+bz|^p\right)^{1/p}$: voir la
  proposition~\ref{tsm}$\,(a)$. Si~$\mu$ est une isom\'etrie, le
  th\'eor\`eme d'\'equimesurabilit\'e de Plotkin-Rudin montre que~$z$
  et~$\th z$ ont m\^eme distribution et donc~$\th^s=1$.
\end{proof}

On peut calculer la norme exacte du multiplicateur de Schur relatif~$\tilde\epsilon$ sur~$\Sch^1_I$ et sur~$\Sch^\infty_I$: elle \'egale la
norme de~$\mu$ sur~$\Ell^1_\Lambda(G)$ et sur~$\Ell^\infty_\Lambda(G)$ avec~$\Lambda=\{1,z\}$ et cette norme est 
\begin{equation*}
  \frac{\max_{z^s=-1}|\vartheta+z|}{|1+\e^{\iu\pi/s}|}
\end{equation*}
(proposition~\ref{cycl}.\ref{cyno}$(d)$ page~\pageref{cyno}.)

Cette proposition est une des \'etapes dans la d\'emonstration du th\'eor\`eme suivant.

\begin{thmf}[page~\pageref{p-np}]
  Soit~$p$ un nombre r\'eel strictement positif qui ne soit pas un entier pair.
  Les propri\'et\'es suivantes sont \'equivalentes.
  \begin{itemize}
  \item $I$~est compl\`etement 1-inconditionnelle complexe dans~$\Sch^p$.
  \item $I$~est 1-inconditionnelle r\'eelle dans~$\Sch^p$.
  \item $I$~est une r\'eunion disjointe d'arbres, c'est-\`a-dire que $I$~ne contient aucun cycle.
  \item Toute suite de signes complexes~$\epsilon\in\T^I$ peut \^etre
    interpol\'ee par une matrice de rang~1.
  \item $I$~est un \emph{ensemble de Varopoulos} de
    \emph{V-interpolation de constante~1:} toute suite~$\varphi\in\ell^\infty_I$ peut \^etre interpol\'ee par un tenseur~$u\in\ell_\Col^\infty\projtens\ell_\Row^\infty$ avec~$\|u\|=\|\varphi\|$.
  \item $I$~est un ensemble d'interpolation isom\'etrique pour les
    multiplicateurs de Schur: toute suite~$\varphi\in\ell^\infty_I$
    est la restriction d'un multiplicateur de Schur sur~$\Sch^\infty$
    de norme~$\|\varphi\|$.
  \end{itemize}
\end{thmf}

Dans le cas o\`u $p$~est un entier pair, la combinatoire devient plus
compliqu\'ee: cela se refl\`ete dans la proposition suivante.

\begin{prp}[page~\pageref{cyno}]
  Si~$p$ est un entier pair, alors $\epsilon$~est un multiplicateur de
  Schur unimodulaire isom\'etrique sur un cycle~$I$ de longueur~$2s$
  pour~$\Sch^p$ si et seulement si~$p/2\in\{1,2,\dots,s-1\}$ ou si~$\epsilon$ peut \^etre interpol\'ee par une matrice de rang~1.
\end{prp}

Cette proposition est une des \'etapes dans la d\'emonstration de la caract\'erisation suivante.

\begin{thmf}[page~\pageref{sigma:THM}]
  Soit~$p$ un entier pair. Les propri\'et\'es suivantes sont
  \'equivalentes.
  \begin{itemize}
  \item $I$~est compl\`etement 1-inconditionnelle complexe dans~$\Sch^p$.
  \item $I$ est 1-inconditionnelle r\'eelle dans~$\Sch^p$.
  \item $I$~ne contient aucun cycle de longueur paire inf\'erieure ou
    \'egale \`a~$p$.
  \end{itemize}
\end{thmf}

Illustrons ce th\'eor\`eme sur l'exemple~\ref{2grb}. Le graphe de
Heawood ne contient aucun cycle de longueur~4: donc la norme~$\Sch^4$ de sa matrice d'incidence ne varie pas si on change le signe
de ses coefficients.

La propri\'et\'e de ne pas contenir de cycle de longueur paire
donn\'ee a \'et\'e tr\`es \'etudi\'ee en th\'eorie des
graphes. Quelle cons\'equence a-t-elle pour la taille du graphe? La
section suivante en propose un r\'esum\'e, \`a comparer aux
r\'esultats du th\'eor\`eme~\ref{densf}.

\section{Graphes bipartis sans cycle de longueur donn\'ee}



\begin{prp}[page~\pageref{extr:prp}]
  Soient~$2\le n\le m$, $I\subseteq\Row\times\Col$ avec~$\card\Col=n$
  et~$\card\Row=m$, et~$e=\card I$.
  \begin{enumerate}
  \item Si $I$~est 1-in\-con\-di\-tio\-nnelle dans~$\Sch^4$ ---~$I$ ne
    contient pas de cycle de longueur~4~--- alors
    \begin{equation*}
      n\ge1 + \Bigl(\frac em-1\Bigr) + \Bigl(\frac
      em-1\Bigr)\Bigl(\frac en-1\Bigr),
    \end{equation*}
    c'est-\`a-dire $e^2-me-mn(n-1)\le0$. On a \'egalit\'e si et
    seulement si $I$ est le graphe d'incidence d'un syst\`eme de
    Steiner $\mathrm S(2,e/m;n)$ sur $n$ points et $m$ blocs (voir
    \cite[Def.~I.3.1]{bjl99} pour la d\'efinition des syst\`emes de
    Steiner.)
  \item Si $I$ est 1-in\-con\-di\-tio\-nnelle dans $\Sch^6$ ---~$I$ ne
    contient pas de cycle de longueur~4 ni~6~--- alors
    \begin{equation*}
      n\ge1 + \Bigl(\frac em-1\Bigr) + \Bigl(\frac
      em-1\Bigr)\Bigl(\frac en-1\Bigr) + \Bigl(\frac
      em-1\Bigr)^2\Bigl(\frac en-1\Bigr),
    \end{equation*}
    c'est-\`a-dire $e^3-(m+n)e^2+2mne-m^2n^2\le0$.  On a \'egalit\'e
    si et seu\-le\-ment si $I$ est le graphe d'incidence du
    \emph{quadrangle} \klammer{le cycle de longueur~8} ou d'un
    quadrangle g\'en\'eralis\'e avec $n$ points et $m$ lignes (voir
    \cite[Def.~1.3.1]{va98} pour la d\'efinition des polygones
    g\'en\'eralis\'es; l'exemple~\ref{2grb} d\'ecrit le plus petit
    quadrangle g\'en\'eralis\'e.)
  \item Si $I$ est 1-in\-con\-di\-tio\-nnelle dans $\Sch^p$ avec
    $p=2k$ un entier pair ---~$I$ ne contient pas de cycle de longueur
    inf\'erieure ou \'egale \`a $p$~--- alors
    \begin{equation}\label{1:hoo}
      n\ge\sum_{i=0}^k\Bigl(\frac em-1\Bigr)^{\lceil\frac i2\rceil}\Bigl(\frac
      en-1\Bigr)^{\lfloor\frac i2\rfloor}.
    \end{equation}
    On a \'egalit\'e si et seulement si $I$ est le graphe d'incidence
    du \emph{$(k+1)$gone} \klammer{le cycle de longueur $2k+2$} ou
    d'un $(k+1)$gone g\'en\'eralis\'e avec $n$ points et $m$ lignes.
  \end{enumerate}
\end{prp}

Les r\'esultats~$(a)$ et~$(b)$ ci-dessus ont \'et\'e obtenus dans la
note \emph{The size of bipartite graphs with girth eight} (voir
pages~\pageref{prp6} et~\pageref{thm:1}), alors que le cas g\'en\'eral
r\'esulte de travaux de Noga Alon, Shlomo Hoory et Nathan Linial (voir
\cite{ho02}).

L'in\'egalit\'e~\eqref{1:hoo} montre que si $I$ est
1-in\-con\-di\-tion\-nelle dans $\Sch^{2k}$, alors $\card I\le
n^{1+1/k}+(s-\nobreak1)n/s$. Si $k\notin\{2,3,5,7\}$, il n'existe pas de
$(k+1)$gones g\'en\'eralis\'es de taille arbitrairement grande et
l'existence de graphes arbitrairement grands sans cycle de longueur
$2k$ de taille minor\'ee par $n^{1+1/k}$ \`a une constante pr\`es est
une question importante en th\'eorie des graphes extr\'emaux. 

La recherche pratique de graphes extr\'emaux nous a amen\'es \`a
\'ecrire un algorithme impl\'ement\'e en langage \texttt{C} qui \'enum\`ere tous
les graphes bipartis d'un nombre de sommets donn\'es et teste
l'existence de cycles. La proposition suivante, d\'emontr\'ee
ind\'ependamment par Adolf Mader et Otto Mutzbauer \cite{mm01},
r\'eduit le nombre de matrices d'incidence de graphes bipartis \`a
tester.

\begin{prp}[page~\pageref{t1}]
  Toute matrice \`a coefficients 0 ou 1 peut \^etre
  simultan\'ement ordonn\'ee selon les ordres lexicographiques des
  lignes et des colonnes (c'est-\`a-dire l'ordre des lignes et des
  colonnes lues comme des nombres binaires) par une permutation des
  lignes et des colonnes.
\end{prp}

En effet, une permutation des lignes et des colonnes de la matrice
d'incidence d'un graphe biparti consiste juste \`a r\'eindexer les
sommets de ce graphe.

\section[Matrices lacunaires et ensembles lacunaires]
{Matrices lacunaires et ensembles lacunaires d'un groupe ab\'elien discret}

\noindent Voici la traduction naturelle entre inconditionnalit\'es de Fourier et matricielle
(les cas $(a)$ et $(b)$ de la section~\ref{sec::1}.)

\begin{prp}[page~\pageref{cor:S}]
  Soit $I\subseteq\Row\times\Col$. Soit $p\in[1,\infty]$: les propri\'et\'es suivantes sont \'equivalentes.
  \begin{itemize}
  \item $I$ est compl\`etement in\-con\-di\-tion\-nelle dans $\Sch^p$.
  \item La suite de produits de Walsh de longueur deux
    $\{\epsilon_c\epsilon'_r:(r,c)\in I\}$ est
    compl\`etement in\-con\-di\-tion\-nelle dans
    $\Ell^p(\{-1,1\}^\Col\times\{-1,1\}^\Row)$.
  \item La suite de produits de deux fonctions de Steinhaus
    $\{z_cz'_r:(r,c)\in I\}$ est compl\`etement
    in\-con\-di\-tion\-nelle dans $\Ell^p(\T^\Col\times\T^\Row)$.
\end{itemize}
 Soit $p\in(0,\infty]$: les propri\'et\'es suivantes sont \'equivalentes.
  \begin{itemize}
  \item $I$ est 1-in\-con\-di\-tion\-nelle dans $\Sch^p$.
  \item La suite $\{\epsilon_c\epsilon'_r:(r,c)\in I\}$ est
    1-in\-con\-di\-tion\-nelle dans
    $\Ell^p(\{-1,1\}^\Col\times\{-1,1\}^\Row)$.
  \item La suite $\{z_cz'_r:(r,c)\in I\}$ est
    1-in\-con\-di\-tion\-nelle dans $\Ell^p(\T^\Col\times\T^\Row)$.
\end{itemize}
\end{prp}
La proposition~\ref{cycl}.\ref{Gamma} (page~\pageref{Gamma}) d\'ecrit
dans quelle mesure cette proposition reste vraie pour d'autres groupes
discrets.

\section{\texorpdfstring{Sous-espaces $\Sch^p_I$ 1-compl\'ement\'es}{Sous-espaces d'entr\'ees 1-compl\'ement\'es dans les classes de Schatten-von-Neumann}}

\noindent En route pour ces r\'esultats, nous obtenons aussi la
caract\'erisation suivante.

\begin{prp}[page~\pageref{complem}]
  Le sous-espace $\Sch^p_I$ de $\Sch^p$ form\'e des op\'erateurs \`a
  support dans $I$ est 1-com\-pl\'e\-men\-t\'e si et seulement si
  $I$ est la r\'eunion disjointe de graphes bipartis complets
  $\Row_j\times\Col_j$: sa matrice d'incidence est, \`a une
  permutation des colonnes et des lignes pr\`es, bloc-diagonale:
\[
\bordermatrix{
  &\scriptstyle C_1&\scriptstyle C_2&\scriptstyle C_3&\scriptstyle\cdots\cr
  \scriptstyle R_1&(1)&(0)&(0)&\cdots\cr
  \scriptstyle R_2&(0)&(1)&(0)&\ddots\cr
  \scriptstyle R_3&(0)&(0)&(1)&\ddots\cr
  \scriptstyle\hfil\vdots&\vdots&\ddots&\ddots&\ddots\cr
  }\text.
\]\end{prp}

\section{\texorpdfstring{Matrices de rang 1 partiellement sp\'ecifi\'ees}{Matrices de rang 1 partiellement sp\'ecifi\'ees}}

\noindent Si $\ph$ est une matrice de rang~1, $\ph=x\tens y$, alors
l'op\'erateur de multiplication de Schur par $\ph$ est de norme
$\sup\abs{x_r}\sup\abs{y_c}$. Or les exemples de calcul exact de normes de
tels op\'erateurs sont tr\`es rares et nous avons voulu savoir comment
cette norme change lorsque $\ph$ agit sur un sous-espace $\Sch^p_I$.
\begin{thmf}[page~\pageref{mps}]
  Soit $I\subset\Row\times\Col$ et consid\'erons
  $(x_\row)_{\row\in\Row}$ et $(y_\col)_{\col\in\Col}$. Alors le
  multiplicateur de Schur relatif $\Sch^p_I$ donn\'e par $(x_\row
  y_\col)_{(\row,\col)\in I}$ est de norme $\sup_{(\row,\col)\in
    I}\abs{x_\row y_\col}$.
\end{thmf}

\section[\texorpdfstring{Propri\'et\'e d'approximation m\'etriquement inconditionnelle dans $\Sch^p_I$}{Propri\'et\'e d'approximation m\'etriquement inconditionnelle pour des sous-espaces d'entr\'ees des classes de Schatten-von-Neumann}]%
{Propri\'et\'e d'approximation m\'etriquement inconditionnelle dans $\Sch^p_I$}

\noindent M\^eme si $I$ n'est pas 1-inconditionnelle dans l'espace $\Sch^p$,
l'espace $\Sch^p_I$ pourrait n\'eanmoins admettre une autre base
1-inconditionnelle. Pour approcher de telles questions, Peter
G.~Casazza et Nigel J.~Kalton ont introduit la propri\'et\'e $(c)$
ci-dessous.
\begin{dfnf}\label{1:muap:def}
  Soit $X$ un espace de Banach s\'eparable et $\Sign=\T$
  (vs.~$\Sign=\{-1,1\}$.)
  \begin{enumerate}
  \item Une suite $(T_k)$ d'op\'erateurs sur $X$ est une \emph{suite
      approximante} si chaque $T_k$ est de rang fini et
    $T_kx\to x$ pour chaque $x\in X$.  
  \item (\cite{pw71}.) Posons $\Delta T_k=T_k-T_{k-1}$. L'espace $X$ a
    la \emph{propri\'et\'e d'approximation inconditionnelle}
    s'il existe une suite
      approximante $(T_k)$ telle que pour une certaine constante~$D$
    \begin{equation*}
      \biggl\|\sum_{k=1}^n\epsilon_k\Delta T_k\biggr\|
      \le D\quad\text{pour tout $n$ et tous $\epsilon_k\in\Sign$}.
    \end{equation*}
    La \emph{constante d'inconditionnalit\'e complexe
      \klammer{vs.~r\'eelle}} de $(T_k)$ est la plus petite des
    constantes~$D$.
  \item (\cite[\S\,3]{ck91}, \cite[\S\,8]{gks93}.) L'espace $X$ a la
    \emph{propri\'et\'e d'approximation m\'etriquement
      inconditionnelle complexe \klammer{vs.~r\'eelle}} si, pour tout
    $\delta>0$, $X$ admet une suite approximante de constante
    d'inconditionnalit\'e complexe \klammer{vs.~r\'eelle} $1+\delta$.
  \end{enumerate}
\end{dfnf}

Voici notre description des sous-espaces $\Sch^p_I$ m\'etriquement inconditionnels:
\begin{thmf}On a deux cas.
  \begin{itemize}
  \item Si $p\in[1,\infty]\setminus\{2,4,6,\dots\}$ et $\Sch^p_I$ a la
    propri\'et\'e d'approximation m\'etriquement inconditionnelle
    r\'eelle, alors la distance d'un sommet colonne \`a un sommet
    colonne est \emph{asymptotiquement} infinie dans $I$: leur
    distance devient arbitrairement grande en effa\c{c}ant un nombre
    fini d'ar\^etes de $I$.
  \item Si $p\in\{2,4,6,\dots\}$, l'espace $\Sch^p_I$ a la
    propri\'et\'e d'approximation m\'etriquement inconditionnelle
    complexe, ou r\'eelle, si et seulement si deux sommets \`a distance
    $2j+1\le p/2$ sont \`a distance asymptotiquement sup\'erieure ou
    \'egale \`a $p-2j+1$.
\end{itemize}
\end{thmf}


\section{In\'egalit\'es matricielles}

\noindent L'article \emph{Matrix inequalities with applications to the theory of
  iterated kernels} montre l'in\'egalit\'e matricielle suivante.
\begin{thmf}[page~\pageref{thm1}]
  Soit $A$ une matrice de taille $n\times m$ \`a coefficients positifs
  et notons $\operatorname{somme}(A)$ la somme de tous ses
  coefficients. On a
  \begin{equation*}
    \operatorname{somme}
    \biggl(\overbrace{AA^*A\dots A\pstar}^{\text{$k$ termes}}\biggr)\ge
    \frac{{\operatorname{somme}(A)}^k}
    {n^{\left\lfloor\frac{k-1}2\right\rfloor}m^{\left\lfloor\frac{k}2\right\rfloor}}
  \end{equation*}
  o\`u $A\pstar$ est $A^*$ ou $A$ selon la parit\'e de $k$.
\end{thmf}

Cette in\'egalit\'e est la version discr\`ete d'un th\'eor\`eme sur
les it\'er\'es d'un noyau (voir la remarque \ref{miwatik}.\ref{noyit}
page~\pageref{noyit}.)  Si on applique cette in\'egalit\'e \`a la
matrice d'incidence d'un graphe biparti $I$, on obtient une minoration
optimale du nombre de promenades de longueur $k$ en termes de la
taille de $I$. Dans le cas $k=3$, une g\'en\'eralisation de cette
in\'egalit\'e (le th\'eor\`eme~\ref{siz}.\ref{AWM} page~\pageref{AWM})
donne une minoration optimale du nombre de chemins (voir le
corollaire~\ref{siz}.\ref{path3} page~\pageref{path3}.)

\section{Transfert entre multiplicateurs de Schur et de Fourier}

\noindent Le th\'eor\`eme suivant est bien connu.
\begin{prp}
  \label{1:trans1}
  Soit $\Gamma$ un groupe discret et $\Row,\Col\subseteq\Gamma$. \`A
  $\Lambda\subset\Gamma$ associons
  $I=\{\,(r,c)\in\Row\times\Col:\row\col\in \Lambda\,\}$. \`A
  $\varrho\in\C^\Lambda$ associons $\varphi\in\C^I$ d\'efini par
  $\varphi(r,c)=\varrho(\row\col)$ pour tout $(r,c)\in I$.
  \begin{itemize}
  \item Soit $p>0$. La norme compl\`ete du multiplicateur de Schur
    relatif $\varphi$ sur $\Sch^p_I$ est born\'ee par la norme
    compl\`ete du multiplicateur de Fourier relatif $\varrho$ sur
    $\Ell^p_\Lambda(\tau)$.
  \item La norme du multiplicateur de Schur relatif $\varphi$ sur
    $\Sch^\infty_I$ est born\'ee par la norme du multiplicateur de
    Fourier relatif $\varrho$ sur $\vn_\Lambda(\tau)$.
  \end{itemize}
\end{prp}

Une forme de r\'eciproque peut \^etre d\'eduite du th\'eor\`eme limite
de Szeg\H{o} matriciel (voir le th\'eor\`eme~\ref{sumsets}.\ref{sz}
page~\pageref{sz}.)
\begin{prp}\label{tsm}
  Soit $\Gamma$ un groupe discret \emph{moyennable} et soit
  $I\subseteq\Gamma\times\Gamma$ un ensemble \emph{toeplitzien} au
   sens que $I=\{(r,c)\in\Gamma\times\Gamma:rc^{-1}\in \Lambda\}$
  pour une partie $\Lambda$ de $\Gamma$. Soit $\varphi\in\C^I$ une
  matrice \emph{toeplitzienne} au le sens qu'il existe
  $\varrho\in\C^\Lambda$ tel que $\varphi(r,c)=\varrho(rc^{-1})$ pour
  tout $(r,c)\in I$.
  \begin{enumerate}
  \item Soit $p>0$. La norme du multiplicateur de Fourier relatif
    $\varrho$ sur $\Ell^p_\Lambda(\tau)$ est le supremum de la norme
    du multiplicateur de Schur relatif $\varphi$ sur des sous-espaces
    de matrices de Toeplitz tronqu\'ees dans $\Sch^p_I$.
  \item De plus, la norme compl\`ete du multiplicateur de Fourier
    relatif $\varrho$ sur $\Ell^p_\Lambda(\tau)$ et la norme
    compl\`ete du multiplicateur de Schur relatif $\varphi$ sur
    $\Sch^p_I$ sont \'egales.
  \item La norme du multiplicateur de Fourier relatif $\varrho$ sur
    $\vn_\Lambda(\tau)$ et la norme du multiplicateur de
    Schur relatif $\varphi$ sur $\Sch^\infty_I$ sont \'egales.
  \end{enumerate}
\end{prp}

\section{Ensembles lacunaires somme de deux ensembles infinis}


\noindent Il est bien connu que les ensembles de Sidon ne peuvent contenir la
somme de deux ensembles infinis; Daniel Li a obtenu la m\^eme
conclusion pour les ensembles $\Lambda$ tels que $\Cont_\Lambda$ admet
une suite approximante m\'etriquement inconditionnelle.  Nous
rassemblons ces deux r\'esultats dans le th\'eor\`eme suivant.
\begin{thmf}
  Soit $\Gamma$ un groupe ab\'elien discret de caract\`eres sur un
  groupe ab\'elien compact $G$. Soit $\Lambda\subset\Gamma$. Si
  $\Gamma$ contient la somme $\Row+\Col$ de deux ensembles infinis
  $\Row$ et $\Col$, alors l'espace $\Cont_\Lambda(G)$ n'admet pas de
  suite approximante inconditionnelle.
\end{thmf}

\begin{proof}[Esquisse de preuve]
  On utilise l'hypoth\`ese pour montrer qu'il existe des parties
  infinies $\Row'$ et $\Col'$ sur lesquelles une \emph{somme \`a blocs
    saut\'es} $\sum(T_{k_{j+1}}-T_{k_j})$ agit comme la projection sur
  la <<~partie triangulaire sup\'erieure~>> de $\Row'+\Col'$. Or ce
  multiplicateur de Fourier relatif se transf\`ere en le
  multiplicateur de Schur qu'est la projection de \emph{Riesz} sur les matrices
  triangulaires sup\'erieures, qui est notoirement non born\'ee.
\end{proof}

Nous obtenons ainsi une preuve \'el\'ementaire que l'alg\`ebre du
disque $\Cont_{\N}(\T)$ n'a pas la propri\'et\'e d'approximation
inconditionnelle, ni l'espace engendr\'e par les fonctions de Walsh de
longueur deux (les produits $\{r_ir_j\}$ de deux fonctions de
Rademacher) dans $\Cont(\{-1,1\}^\infty)$, ni l'espace engendr\'e par
les produits $\{s_is_j\}$ de deux fonctions de Steinhaus dans
$\Cont({\T}^\infty)$.

Nous montrons aussi qu'un ensemble <<~compl\`etement $\Lambda(p)$~>>
ne peut contenir la somme de deux ensembles infinis
(th\'eor\`eme~\ref{sumsets}.\ref{har:lp} de la page~\pageref{har:lp}.)  

La preuve ci-dessus montre aussi que la constante
d'inconditionnalit\'e r\'eelle pour les espaces $\Ell^p_\Lambda(G)$
est minor\'ee par la norme compl\`ete de la projection de Riesz sur $S^p$. Cela nous a motiv\'es pour
calculer cette norme et, \`a d\'efaut, la norme compl\`ete de la
transformation de Hilbert matricielle.
\begin{thmf}
  La norme compl\`ete de la projection de Riesz et de la
  transformation de Hilbert matricielle sur $\Sch^p$ co\"{\i}ncident
  avec leur norme.  
  \begin{itemize}
  \item Si $p$ est un entier pair, la norme de la transformation de
    Hilbert matricielle est $\cot(\pi/2p)$ (voir
    page~\pageref{normh}).
  \item La norme de la projection de Riesz sur $\Sch^4$ est $\sqrt2$ (voir page~\pageref{normt}).
  \end{itemize}
\end{thmf}

\section[Probl\`emes extr\'emaux pour les polyn\^omes
trigonom\'etriques]{Probl\`emes extr\'emaux pour les polyn\^omes trigonom\'etriques}


\noindent Cette th\'ematique de recherche correspond au cas de figure
$(a)$ de la question~\ref{q:inc}, avec $\R/\Z$ comme groupe ab\'elien
et le module maximum comme norme.

Soit $\Lambda=\{\lambda_1,\lambda_2,\dots,\lambda_n\}$ un ensemble de
$n$ entiers. Pour $n$ nombres r\'eels positifs $r_1,r_2,\dots,r_n$ et
$n$ nombres r\'eels $t_1,t_2,\dots,t_n$, consid\'erons le
polyn\^ome trigonom\'etrique
\begin{equation}
  \label{tti}
  f(x)=r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+\dots+r_n\ei{(t_n+\la_nx)}.
\end{equation}
La question de l'inconditionnalit\'e dans l'espace $\Cont$ des
fonctions continues est alors celle de la d\'ependance du module
maximum du polyn\^ome trigonom\'etrique $f$ par rapport aux arguments
(phases) $t_1,\allowbreak t_2,\allowbreak \dots,\allowbreak t_n$.

Voici quatre probl\`emes qui \'eclairent divers aspects de
l'inconditionnalit\'e.

\begin{epf}[probl\`eme de Man\-del$'$\-shtam complexe ---~voir
  \protect{\cite[page~2 et le suppl\'ement]{dm90}}]
  \label{pe1}
  Trouver le minimum du module maximum du polyn\^ome trigonometrique
  $f$ pour des modules de coefficients de Fourier $r_1,r_2,\dots,r_n$
  donn\'es:
\begin{equation*}
  \min_{t_1,t_2,\dots,t_n}\max_x|r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+\dots+r_n\ei{(t_n+\la_nx)}|.
\end{equation*}
\end{epf}
\begin{epf}
  \label{pe2}
  Trouver le minimum du module maximum du polyn\^ome trigonometrique
  $f$ pour un spectre $\Lambda$, des arguments $t_1,t_2,\dots,t_n$ et
  la somme des modules $r_1+r_2+\dots+r_n$ des coefficients de Fourier
  donn\'es:
  \begin{equation*}
  \min_{r_1,r_2,\dots,r_n}\max_x
  \frac{\abs{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+\dots+r_n\ei{(t_n+\la_nx)}}}
  {r_1+r_2+\dots+r_n}.
\end{equation*}
\end{epf}
\begin{pf}
  \label{normul}
  Trouver le maximum de la variation du module maximum du polyn\^ome
  trigonometrique $f$ pour un spectre $\Lambda$ et une variation des
  arguments $\Delta t_1,\Delta t_2,\dots,\Delta t_n$ donn\'ee. En
  d'autres termes, trouver la norme du multiplicateur de Fourier
  relatif unimodulaire par les signes $\ei{\Delta t_1},\allowbreak\ei{\Delta t_2},\allowbreak\dots,\allowbreak\ei{\Delta t_n}$:
\vskip\abovedisplayskip\centerline{$\displaystyle
  \min_{\substack{r_1,r_2,\dots,r_n\\t_1,t_2,\dots,t_n}}\frac
  {\max_x\abs{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+\dots+r_n\ei{(t_n+\la_nx)}}}
  {\max_x\abs{r_1\ei{(t_1+\Delta t_1+\la_1x)}+r_2\ei{(t_2+\Delta t_2+\la_2x)}+\dots+r_n\ei{(t_n+\Delta t_n+\la_nx)}}}.$}
\vskip\belowdisplayskip
\end{pf}
\begin{epf}[constante de Sidon]
  \label{pe3}
  Trouver le minimum du module maximum du polyn\^ome trigonometrique
  $f$ pour un spectre $\Lambda$ et la somme des modules
  $r_1+r_2+\dots+r_n$ des coefficients de Fourier donn\'es:
  \begin{equation*}
  \min_{\substack{r_1,r_2,\dots,r_n\\t_1,t_2,\dots,t_n}}
  \max_x\frac{\abs{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+\dots+r_n\ei{(t_n+\la_nx)})}}
  {r_1+r_2+\dots+r_n}.
\end{equation*}
L'inverse de ce minimum est la constante d'inconditionnalit\'e de
$\Lambda$ dans l'espace des fonctions continues: c'est la \emph{constante de
Sidon} de $\Lambda$.
\end{epf}


Littlewood~\cite{li24} et Salem~\cite{sa33} se sont int\'eress\'es \`a
ces probl\`emes. Ils sont aussi apparus dans la th\'eorie du circuit
\'electrique, comme le raconte N.~G.  Chebotar\"ev: <<~L.~I.
Man\-del$'$\-shtam m'a communiqu\'e un probl\`eme sur le choix des
phases de courants \'electriques de fr\'equences diff\'erentes de
sorte que la capacit\'e du courant r\'esultant de faire sauter les
fusibles soit minimal~>> \cite[p.~396]{ce49}. Ce probl\`eme est une de
ses motivations pour proposer une formule pour la valeur des
d{\'e}riv{\'e}es directionnelles d'une fonction maximum en fonction
des d{\'e}riv{\'e}es des fonctions dont on prend le maximum, qui a
\'et\'e un peu oubli\'ee malgr\'e son caract\`ere naturel.

\begin{Chf}
  [\protect{\cite[Theorem~VI.3.2, (3.6)]{dm90}}] Soit $I$ un ouvert de
  $\R^n$ et soit $K$ un espace compact. Soit $F(t,x)$ une fonction
  r\'eelle sur $I\times K$ qui soit continue, ainsi que
  $\dfrac{\partial F}{\partial t}(t,x)$. Soit
  \begin{equation*}
    F^*(t)=\max_{x\in K}F(t,x).
  \end{equation*}
  Alors $F^*(t)$ admet le d\'eveloppement limit\'e suivant en tout
  $t\in I$:
  \begin{equation}
    \label{forcheb}
    F^*(t+h)
    =F^*(t)
    +\max_{F(t,x)=F^*(t)}
    \Bigl\langle{h,\dfrac{\partial F}{\partial t}(t,x)}\Bigr\rangle
    +o(h).  
  \end{equation}
\end{Chf}
Chebotar\"ev utilise en particulier cette formule pour r\'esoudre le
probl\`eme d'approximation polynomiale de Chebyshev et le probl\`eme
du minimum d'une forme quadratique sur les entiers de Korkin et
Zolotarev.


\section[Probl\`emes extr\'emaux pour les trin\^omes
trigonom\'etriques]{Probl\`emes extr\'emaux pour les trin\^omes trigonom\'etriques}

\noindent Ces probl\`emes sont d\'ej\`a int\'eressants dans le cas
$n=3$. Dans l'article \emph{The Sidon constant of sets with three elements},
nous avons r\'esolu les probl\`emes extr\'emaux~\ref{pe1} et~\ref{pe3}
pour ce cas. Nous allons supposer, en toute g\'en\'eralit\'e, que
$\Lambda$ est un ensemble de trois entiers $\la_1<\la_2<\la_3$ tels
que $\la_2-\la_1$ et $\la_3-\la_2$ sont premiers entre eux. 

Il s'av\`ere que les arguments $t_1,t_2,t_3$ d'un trin\^ome
trigonom\'etrique
  \begin{equation}\label{TTi}
    f(x)=r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}
  \end{equation}
  donnent lieu \`a un param\`etre unique, que nous appellerons
  l'\emph{argument du trin\^ome} $f$: la d\'etermination principale
  $\tau\in\iog[-\pi,\pi]$ de
\begin{equation}\label{argtri}
  (\la_3-\la_2)t_1+
  (\la_1-\la_3)t_2+
  (\la_2-\la_1)t_3\bmod{2\pi\Z}.
\end{equation}

Nous avons ainsi \'etabli que les arguments minimaux du probl\`eme
extr\'emal~\ref{pe1} correspondent \`a des multiples de $\pi$:

\begin{thmf}[page~\pageref{spe1}]
  Soit $\Lambda$ un ensemble de trois entiers. Soient $r_1$, $r_2$ et $r_3$
  trois r\'eels strictement positifs. Les arguments $t_1,t_2,t_3$
  r\'esolvent le probl\`eme extr\'emal~\ref{pe1} si et seulement si
  l'argument $\tau$ du trin\^ome \'egale $\pi$. En particulier,
  $t_1$, $t_2$ et $t_3$ peuvent \^etre choisis parmi 0 et $\pi$.
\end{thmf}

Ce th\'eor\`eme permet de  d\'eterminer les coefficients de
Fourier minimaux pour le probl\`eme extr\'emal~\ref{pe3}:
\begin{prp}[page~\pageref{spe3}]
  Le polyn\^ome suivant r\'esout le probl\`eme extr\'emal~\ref{pe3}:
  \begin{equation*}
    \label{fspe3}
    f(x)=
    \epsilon_1(\la_3-\la_2)\ei{\la_1x}+
    \epsilon_2(\la_3-\la_1)\ei{\la_2x}+
    \epsilon_3(\la_2-\la_1)\ei{\la_3x}
  \end{equation*}
o\`u $\epsilon_1,\epsilon_2,\epsilon_3\in\{-1,1\}$ sont trois signes r\'eels tels que
\begin{itemize}
\item $\epsilon_1\epsilon_2=-1$ si $\la_2-\la_1$ est pair;
\item $\epsilon_1\epsilon_3=-1$ si $\la_3-\la_1$ est pair;
\item $\epsilon_2\epsilon_3=-1$ si $\la_3-\la_2$ est pair.
\end{itemize}
La constante de Sidon de $\Lambda$ \'egale donc
${\cos(\pi/2(\la_3-\la_1))}^{-1}$ et les constantes
d'inconditionnalit\'e complexes et r\'eelles de $\Lambda$ dans
l'espace des fonctions continues co\"{\i}ncident donc pour les
ensembles $\Lambda$ \`a trois \'el\'ements.
\end{prp}
Nous avions entam\'e cette direction de recherche pour v\'erifier que
les constantes d'inconditionnalit\'e complexes et r\'eelles d'un
ensemble $\Lambda$ \'etaient bien diff\'erentes; les ensembles \`a
trois \'el\'ements ne fourniront pas de contre-exemple et la question
demeure ouverte.

En fait, les probl\`emes extr\'emaux~\ref{pe2} et~\ref{pe3}
admettent une solution \'el\'ementaire: on ram\`ene le trin\^ome
trigonom\'etrique \`a la forme <<~normale~>>
\begin{equation}\label{fnspe3}
    r_1\emi{kx}+r_2\ei{\tau/(k+l)}+r_3\ei{lx}
\end{equation}
avec $k$ et $l$ positifs et premiers entre eux et $\tau\in\iog[-\pi,\pi]$. Alors 
  \begin{align*}
    \frac{\max_x\bigabs{r_1\emi{kx}+r_2\ei{\tau/(k+l)}+r_3\ei{lx}}}{r_1+r_2+r_3}
    &\ge\frac{\bigabs{r_1+r_2\ei{\tau/(k+l)}+r_3}}{r_1+r_2+r_3}\\
    &=\sqrt{1-\frac{4(r_1+r_3)r_2}{{(r_1+r_2+r_3)}^2}\sin^2\frac\tau{2(k+l)}}\\
    &\ge\sqrt{1-\sin^2\frac\tau{2(k+l)}}=\cos\frac\tau{2(k+l)},
  \end{align*}
  et si $r_1:r_2:r_3=l:k+l:k$, le trin\^ome
  trigonom\'etrique~\eqref{fnspe3} atteint son module maximum en~0
  et satisfait $r_1+r_3=r_2$.
%\section{Les points de module maximum d'un trin\^ome trigonom\'etrique}

\section[\texorpdfstring{Points extr\'emaux et expos\'es de la boule unit\'e de
  l'espace $\Cont_\Lambda$}{Points extr\'emaux et expos\'es de la boule unit\'e d'un espace de trin\^omes trigonom\'etriques}]%
{Points extr\'emaux et points expos\'es de la boule unit\'e de
  l'espace $\Cont_\Lambda$}

\noindent Si on cherche \`a r\'esoudre les probl\`emes
extr\'emaux~\ref{pe1},~\ref{pe2} et~\ref{pe3} par une application de
la formule de Chebotar\"ev~\eqref{forcheb}, il est utile d'obtenir des
informations sur les points $x$ tels que <<~$F(t,x)=F^*(t)$,~>>
c'est-\`a-dire les points maximum de $F(t,\cdot)=\abs{f}$. Par
exemple, on peut d\'eduire de cette formule qu'il y en a plus d'un car
sinon $(t,x)$ serait un point critique de $F$; or un petit calcul
(lemme~\ref{scis}.\ref{ubc:lem} page~\pageref{ubc:lem}) montre que ce n'est pas
possible.

Voici un argument d'analyse fonctionnelle qui d\'emontre la m\^eme
chose. Comme les probl\`emes ci-dessus sont lin\'eaires, on peut
limiter la recherche de polyn\^omes trigonom\'etriques extr\'emaux aux
points expos\'es de la boule unit\'e $K$ de l'espace $\Cont_\Lambda$ 
(rappelons qu'un point $P$ de $K$ est expos\'e par un hyperplan $H$ si
$H$ ne coupe $K$ qu'en $P$.) Pourquoi un point expos\'e $P$ de $K$
atteint-il son module maximum en au moins deux points? parce que la
forme lin\'eaire qui d\'efinit l'hyperplan $H$ s'\'etend en une mesure
$\mu$ qui atteint sa norme sur $P$ et on sait que $P$ doit \^etre de
module maximum sur le support de $\mu$; la mesure $\mu$ n'est pas une
masse de Dirac puisqu'elle atteint sa norme uniquement en $P$, de
sorte que le support de $\mu$ a au moins deux points.

Dans l'article \emph{The maximum modulus of a trigonometric
  trinomial}, nous obtenons une description tr\`es compl\`ete des
points de module maximum d'un trin\^ome trigonom\'etrique (voir le
th\'eor\`eme~\ref{tmmoatt}.\ref{mmpttt} page~\pageref{mmpttt}) dont voici le point
saillant.
\begin{thmf}
  Soit $\Lambda$ un ensemble de trois entiers $\la_1<\la_2<\la_3$ tels
  que $\la_2-\la_1$ et $\la_3-\la_2$ soient premiers entre eux. Soient
  $r_1,r_2,r_3$ trois r\'eels strictement positifs. Le trin\^ome
  trigonom\'etrique
  \begin{equation*}
    f(x)=r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}
  \end{equation*}
  atteint son module maximum en un point unique modulo $2\pi$, de
  multiplicit\'e 2, sauf si son argument $\tau$ \'egale $\pi$: Si $f$
  atteint son module maximum en deux points modulo $2\pi$, c'est
  parce que son graphe admet un axe de sym\'etrie.
\end{thmf}

\begin{proof}[Esquisse de d\'emonstration]
  On ram\`ene le trin\^ome trigonom\'etrique $f$ \`a la forme
  normale~\eqref{fnspe3} avec de plus $\tau\in[0,\pi]$ et on montre
  alors que $f$ doit atteindre son module maximum sur le petit
  intervalle $[-\tau/k(k+l),\tau/l(k+l)]$ en trouvant, pour tout $y$
  hors de cet intervalle, un point $x$ qui y soit pour lequel
  $\abs{f(x)}\ge\abs{f(y)}$. De plus, on peut rendre cette
  in\'egalit\'e stricte sauf si $\tau=\pi$. Il reste alors \`a
  \'etudier les variations de $\abs{f}$ sur
  $[-\tau/k(k+l),\tau/l(k+l)]$.
\end{proof}

La formule de Chebotar\"ev donne alors une nouvelle solution pour le
probl\`eme extr\'emal~\ref{pe1}.
\begin{prp}
  Le module maximum de $r_1\emi{kx}+r_2\ei{\tau/(k+l)}+r_3\ei{lx}$ est
  une fonction strictement d\'ecroissante de $\tau$ sur $[0,\pi]$.
\end{prp}

\begin{proof}
Restons dans le contexte de l'esquisse de d\'emonstration ci-dessus et soit $\tau\in\io[0,\pi]$. Soit $x^*$ l'unique point de module maximum pour $f$: on a vu que $x^*\in[-\tau/k(k+l),\tau/l(k+l)]$. Mais alors
\begin{multline*}
    \abs{f(x)}^2=r_1^2+r_2^2+r_3^2+2r_1r_3\cos\bigl((k+l)x\bigr)
    +2r_2\bigl(r_1\cos(\tau/(k+l)+kx)+r_3\cos(\tau/(k+l)-lx)\bigr)
  \end{multline*}
et
\begin{equation*}
      \frac{k+l}{2r_2}\dfrac{\partial\abs{f}^2}{\partial\tau}(x^*)
    =-r_1\sin(\tau/(k+l)+kx^*)-r_3\sin(\tau/(k+l)-lx^*)<0
\end{equation*}
car $\tau/(k+l)+kx^*\in[0,\tau/l]$ et $\tau/(k+l)-lx^*\in[0,\tau/k]$
ne s'annulent pas simultan\'emanent.
\end{proof}

Illustrons notre propos: le module maximum de
$f(x)=4\mkern1mu\emi{2x}+\ei{t}+\ei{x}$ est la distance maximum de
points de l'hypotrocho\"{\i}de d'\'equation
$z=4\mkern1mu\emi{2x}+\ei{x}$ \`a un point donn\'e $-\ei{t}$ du plan
complexe. Nous avons donc montr\'e que si deux points de $H$ sont
simultan\'ement \`a distance maximum de $-\ei{t}$, alors $-\ei{t}$ est
sur un axe de sym\'etrie de $H$, c'est-\`a-dire
$t\equiv\pi/3\bmod{2\pi/3}$. \vskip\abovedisplayskip
\hfil\begin{pspicture}(-2.6875,-3.14805)(3.33333,3.14805)
  \parametricplot[plotstyle=ccurve,linewidth=.4pt]{0}{360}{t cos 2 mul
    3 div t 2 mul cos 8 mul 3 div add t sin 2 mul 3 div t 2 mul sin 8
    mul 3 div sub}
  \pscircle[linewidth=.4pt](0,0){.66667}
  \psline[linewidth=.7pt,linestyle=dotted,dotsep=1.5pt](-.66667,0)(3.33333,0)
  \psline[linewidth=.4pt,linestyle=dashed,dash=2pt 2pt](-1.08152,
  3.10770)(-.33333,-.57735)(3.23211,.61723) \rput(-0.90,0){\small$-1$}
  \rput(-.65,-.80){\small$-\ei{\pi/3}$}
\end{pspicture}\vskip\belowdisplayskip

Notre \'etude aboutit au th\'eor\`eme suivant, dont on peut esp\'erer
une g\'en\'eralisation \`a des ensembles $\Lambda$ plus grands.

\begin{thmf}[page~\pageref{extrpts}]
  Soit $\Lambda$ un ensemble \`a trois \'el\'ements. Soit $K$ la boule
  unit\'e de l'espace $\Cont_\Lambda$ et soit $P\in K$.
  \begin{itemize}
  \item Le point $P$ est un point expos\'e de $K$ si et seulement si
    $P$ est un mon\^ome trigonom\'etrique $\ei\alpha\ei{\la x}$ avec
    $\alpha\in\R$ et $\la\in\Lambda$ ou un trin\^ome trigonom\'etrique
    qui atteint son module maximum, 1, en deux points modulo
    $2\pi/d$. Toute forme lin\'eaire sur $\Cont_\Lambda$ atteint sa
    norme en un point expos\'e de $K$.
  \item Le point $P$ est un point extr\'emal de $K$ si et seulement si
    $P$ est un mon\^ome trigonom\'etrique $\ei\alpha\ei{\la x}$ avec
    $\alpha\in\R$ et $\la\in\Lambda$ ou un trin\^ome trigonom\'etrique
    tel que $1-{\abs P}^2$ a quatre z\'eros modulo $2\pi$, compt\'es
    avec leur multiplicit\'e.
  \end{itemize}
\end{thmf}


\section{La variation du module maximum en fonction de l'argument}

Nous utilisons aussi la formule de Chebotar\"ev pour montrer que le
module maximum d'un trin\^ome trigonom\'etrique est une fonction
d\'ecroissante de la valeur absolue $\abs\tau$ de son argument
(voir~\eqref{argtri}) et pour borner cette d\'ecroissance. Nous
obtenons les in\'egalit\'es suivantes.
\begin{thmf}[pages~\pageref{how-dec},~\pageref{th-4} et ~\pageref{multiplier}]
  Soit $f$ un trin\^ome trigonom\'etrique comme en~\eqref{TTi} et
  varions ses arguments de $\Delta t_1,\Delta t_2,\Delta t_3$. Notons
  $\tilde f$ le trin\^ome qui en r\'esulte, $\tilde\tau$ l'argument du
  trin\^ome $\tilde f$, et $\Delta\tau$ la variation de l'argument:
  \begin{equation*}
    \Delta\tau
    \equiv(\la_3-\la_2)\Delta t_1
    +(\la_1-\la_3)\Delta t_2
    +(\la_2-\la_1)\Delta t_3\mod{2\pi\Z}.
  \end{equation*}
  Si $\abs{\tilde\tau}>\abs\tau$, alors
  \begin{align}
    {\max_x\abs{\tilde f(x)}}
    &<{\max_x\abs{f(x)}}\notag\\
    &\le\frac{\abs{r_1+r_2\ei{\tau/\abs{\la_3-\la_1}}+r_3}}
    {\abs{r_1+r_2\ei{\tau\abs{\la_3-\la_1}}+r_3}}{\max_x\abs{\tilde f(x)}}\label{pi1}\\
    &\le\frac{\cos(\tau /2(\la_3-\la_1))}{\cos(\tilde\tau/2(\la_3-\la_1))}
    {\max_x\abs{\tilde f(x)}}\label{pi2}\\
    &\le\frac{\cos((\pi-\abs{\Delta\tau})/2(\la_3-\la_1))}{\cos(\pi/2(\la_3-\la_1))}
    {\max_x\abs{\tilde f(x)}};\label{nmfu}
  \end{align}
  l'in\'egalit\'e~\eqref{pi1} est une \'egalit\'e si et seulement si
  $r_1:r_3=\la_3-\la_2:\la_2-\la_1$, l'in\'egalit\'e~\eqref{pi2} si et
  seulement si $r_1:r_2:r_3=\la_3-\la_2:\la_3-\la_1:\la_2-\la_1$ et
  l'in\'egalit\'e~\eqref{nmfu} si et seulement si de plus
  $\tau=\pi$. La norme du multiplicateur de Fourier relatif
  unimodulaire par les signes $\ei{\Delta t_1},\ei{\Delta
    t_2},\dots,\ei{\Delta t_n}$ est donc le facteur dans
  l'in\'egalit\'e~\eqref{nmfu}.
\end{thmf}

En particulier, le module maximum de $f$ comme fonction de $\tau$
admet les deux minorants
$\abs{r_1+r_2\ei{\tau/\abs{\la_3-\la_1}}+r_3}$ et
$(r_1+r_2+r_3)\cos(\tau /2(\la_3-\la_1))$ sur l'intervalle $[-\pi,\pi]$. Illustrons-le dans le cas particulier
$f(x)=4\emi{2x}+\ei{\tau/3}+\ei{x}$:\vskip\abovedisplayskip
{\psset{xunit=3cm,yunit=3cm,linewidth=.4pt}
\begin{pspicture}(-1.6,4.85)(1.6,6.1) 
    \savedata{\V}[-.20944 5.9852 -.15708 5.9917 -.10472 5.9963
      -.052361 5.9991 0 6 .052361 5.9991 .10472 5.9963 .15708 5.9917
      .20944 5.9852 .2618 5.9768 .31416 5.9666 .36653 5.9546 .41887
      5.9407 .47124 5.9251 .52361 5.9077 .57595 5.8885 .62832 5.8676
      .68069 5.8449 .73303 5.8206 .7854 5.7946 .83777 5.7669 .89011
      5.7376 .94248 5.7068 .99485 5.6743 1.0472 5.6404]
    \dataplot[plotstyle=curve]{\V}
    \savedata{\U}[1.0472 5.6404 1.0996 5.6743 1.1519 5.7068 1.2043
      5.7376 1.2566 5.7669 1.309 5.7946 1.3613 5.8206 1.4137 5.8449
      1.4661 5.8676 1.5184 5.8885 1.5708 5.9077]
    \dataplot[plotstyle=curve]{\U}
    \psline[linestyle=solid]{->}(0,4.95)(0,6.1)
    \psline[linestyle=dashed]{->}(-0.20944,5)(1.6,5)
    \rput(0,4.88){\small0} \rput[b](0,6.12){\small$m$}
    \rput[r](-.23,5){\small5} \rput[r](1.66,5){\small$\tau$}
    \rput(1.0472,4.88){\small$\pi$}
    \rput[l](-1.7,5.97){\small$m=\max\limits_x|4\emi{2x}+\ei{\tau/3}+\ei{x}|$}
    \rput[l](1.07,5.57){\small$m=|4+\ei{\tau/3}+1|$}
    \rput[l](1.07,5.2){\small$m=(4+1+1)\cos(\tau/6)$}
    \psline[linestyle=dashed](1.0472,4.95)(1.0472,6.1)
    \psplot[plotstyle=curve,linestyle=dashed,dash=2pt
    2pt]{0}{1.0472}{x 57.29578 mul cos 10 mul 26 add sqrt}
    \psplot[plotstyle=curve,linestyle=dotted,linewidth=.7pt,dotsep=1.5pt]{0}{1.0472}{x
      57.29578 mul 2 div cos 6 mul}
  \end{pspicture}}
 

\section[Probl\`emes extr\'emaux pour les quadrin\^omes
trigonom\'etriques]%
{Probl\`emes extr\'emaux pour les quadrin\^omes trigonom\'etriques}
\noindent Dans le cas $\Lambda=\{0,1,2,3\}$, le probl\`eme
extr\'emal~\ref{pe3} est un probl\`eme ouvert pos\'e par Harold
S.~Shapiro en 1951.  Par des moyens heuristiques, nous avons
conjectur\'e que les polyn\^omes extr\'emaux sont de la forme
\vskip\abovedisplayskip \centerline{$\displaystyle
  \frac{\iu2\sqrt2\cos u-1-3\sin u}{15} + \frac{3+\sin u}{10}\e^{\iu x
  }+ \frac{3-\sin u}{10}\e^{\iu2x} + \frac{\iu2\sqrt2\cos u-1+3\sin
    u}{15}\e^{\iu3x}$}\vskip\belowdisplayskip
\noindent o\`u $u$ parcourt $[0,2\pi\mathclose[$. Ces polyn\^omes sont
\'etudi\'es dans la note \emph{On the Sidon constant of
  $\{0,1,2,3\}$}, section~\ref{chquatre}.\ref{distpol}. On en
d\'eduirait que la constante de Sidon de $\Lambda$ vaut $5/3$, qui est
sa constante d'inconditionnalit\'e r\'eelle (voir la
proposition~\ref{chquatre}.\ref{cir} page~\pageref{cir}.)  Il s'agit
de le d\'emontrer et d'\'etudier plus g\'en\'eralement les polyn\^omes
trigonom\'etriques \`a quatre termes dont le module atteint son
maximum en trois points.\clearpage

\selectlanguage{british}

\chapter{Lacunary matrices} 
\label{lac}
\begin{flushright}
  \large with Asma Harcharras and Krzysztof Oleszkiewicz.
\end{flushright}\smallskip

\begin{abstract}
%
  \noindent
  We study unconditional subsequences of the canonical basis
  $(\e_{rc})$ of elementary matrices in the Schatten class $S^p$. They
  form the matrix counterpart to Rudin's $\Lambda(p)$ sets of integers
  in Fourier analysis.  In the case of $p$ an even integer, we find a
  sufficient condition in terms of trails on a bipartite graph. We
  also establish an optimal density condition and present a random
  construction of bipartite graphs. As a byproduct, we get a new proof
  for a theorem of Erd\H os on circuits in graphs.
%
\end{abstract}
%
\section{Introduction}

\noindent We study the following question on the Schatten class
$S^p$.\vskip\abovedisplayskip

\block{$(\dagger)$}{How many matrix coefficients of an operator $x\in
  S^p$ must vanish so that the norm of $x$ has a bounded variation if
  we change the sign of the remaining nonzero matrix
  coefficients?}\vskip\belowdisplayskip

Let $\Col$ be the set of columns and $\Row$ be the set of rows for
coordinates in the matrix, in general two copies of $\N$. Let
$I\subseteq \Row\times \Col$ be the set of matrix coordinates of the
remaining nonzero matrix coefficients of $x$. Property $(\dagger)$
means that the subsequence $(\e_{rc})_{(r,c)\in I}$ of the canonical
basis of elementary matrices is an unconditional basic sequence in
$S^p$: $I$ forms a $\sigma(p)$ set in the terminology of
\cite[\S4]{ha98}.

It is natural to wonder about the operator valued case, where the
matrix coefficients are themselves operators in $S^p$. As the proof of
our main result carries over to that case, we shall state it in the
more general terms of complete $\sigma(p)$ sets.

We show that for our purpose, a set of matrix entries $I\subseteq
\Row\times \Col$ is best understood as a bipartite graph. Its two
vertex classes are
% set $V$ is the disjoint union of
$\Col$ and $\Row$, whose elements will respectively be termed ``column
vertices'' and ``row vertices''. Its edges join only row vertices
$r\in \Row$ with column vertices $c\in \Col$, this occurring exactly
if $(r,c)\in I$.
% , i.e., $I$ is the incidence matrix of this graph.

We obtain a generic condition for $\sigma(p)$ sets in the case of even
$p$ (Theorem~\ref{sig:thm}) that generalises
\cite[Prop.~6.5]{ha98}. These sets reveal in fact as a matrix
counterpart to Rudin's $\Lambda(p)$ sets and we are able to transfer
Rudin's proof of \cite[Theorem~4.5(b)]{ru60} to a non-commutative
context: his number $r_s(E,n)$ is replaced by the numbers of
Def.~\ref{def2}$(b)$ and we count trails between given vertices
instead of representations of an integer.

We also establish an upper bound for the intersection of a $\sigma(p)$
set with a finite product set $\Row'\times \Col'$ (Theorem~\ref{dc}):
this is a matrix counterpart to Rudin's \cite[Theorem~3.5]{ru60}. In
terms of bipartite graphs, this intersection is the subgraph induced
by the vertex subclasses $\Col'\subseteq\Col$ and
$\Row'\subseteq\Row$.

The bound of Theorem~\ref{dc} provides together with
Theorem~\ref{sig:thm} a generalisation of a theorem by Erd\H os
\cite[p.~33]{er64} on graphs without circuits of a given even
length. In the last part of this article, we present a random
construction of maximal $\sigma(p)$ sets for even integers $p$.


\begin{Ter}
  $\Col$ is the set of columns and $\Row$ is the set of rows, in
  general both indexed by $\N$. The set $V$ of all vertices is their
  disjoint union $\Row\amalg \Col$. An edge on $V$ is a pair
  $\{v,w\}\subseteq V$. A graph on $V$ is given by its set of edges
  $E$. A bipartite graph on $V$ with vertex classes $\Col$ and $\Row$
  has only edges $\{r,c\}$ such that $c\in\Col$ and $r\in\Row$ and may
  therefore be described alternatively by the set
  $I=\{\,(r,c)\in\Row\times\Col:\{r,c\}\in E\,\}$. A trail of length $s$
  in a graph is a sequence $(v_0,\dots,v_s)$ of $s+1$ vertices such
  that $\{v_0,v_1\},\dots,\{v_{s-1},v_s\}$ are pairwise distinct edges
  of the graph. A trail is a path if its vertices are pairwise
  distinct. A circuit of length $p$ in a graph is a sequence
  $(v_1,\dots,v_p)$ of $p$ vertices such that
  $\{v_1,v_2\},\dots,\{v_{p-1},v_p\},\{v_p,v_1\}$ are pairwise
  distinct edges of the graph. A circuit is a cycle if its vertices
  are pairwise distinct.
\end{Ter}

\begin{Not}
  $\T=\{\,z\in\C:|z|=1\,\}$. Let $q=(r,c)\in \Row\times \Col$. The
  transpose of $q$ is $q^*=(c,r)$. The entry\index{entry} (elementary
  matrix) $\e_q=\e_{rc}$ is the operator on $\ell_2$ that maps the
  $c$th basis vector on the $r$th basis vector and all other basis
  vectors on 0.  The matrix coefficient at coordinate $q$ of an
  operator $x$ on $\ell_2$ is $x_q=\tr\e_q^*x$ and its matrix
  representation is $(x_q)_{q\in \Row\times \Col}=\sum_{q\in
    \Row\times \Col}x_q\e_q$.  The Schatten class $S^p$, $1\le
  p<\infty$, is the space of those compact operators $x$ on $\ell_2$
  such that $\|x\|_p^p=\tr|x|^p=\tr(x^*x)^{p/2}<\infty$.  For
  $I\subseteq \Row\times \Col$, the entry space\index{entry subspace
    of $S^p$} $S^p_I$ is the space of those $x\in S^p$ whose matrix
  representation is supported by $I$: $x_q=0$ if $q\notin I$. $S^p_I$
  is also the closed subspace of $S^p$ spanned by $(\e_q)_{q\in I}$.
  % The operator valued entry space $S^p_I(S^p)$ is the closed
  % subspace of $S^p(S^p)\equiv S^p(\ell_2\otimes_2\ell_2)$, endowed
  % with $\|x\|_p^p=\tr\otimes\tr|x|^p$, that is spanned by the
  % $x_q\e_q$ with $x_q\in S^p$ and $q\in I$. One should rather write
  % $x_q\otimes\e_q$ instead of $x_q\e_q$, but we think of
  % $x_q=\Id_{S^p}\otimes\tr((\Id_{\ell_2}\otimes\e_q^*)x)$ as the
  % operator coefficient of $x$ at matrix coordinate $q$.  Thus, for
  % even integer $p$ and $x=(x_q)_{q\in I}=\sum_{q\in I}x_q\e_q$
  % $(=\sum_{q\in I}x_q\otimes\e_q)$ with $x_q\in S^p$ and $I$
  % finite, $$ \|x\|_p^p=\sum_{q_1,\dots q_p\in I}\tr
  % x_{q_1}^*x_{q_2}\dots x_{q_{p-1}}^*x_{q_p}\tr
  % \e_{q_1}^*\e_{q_2}\dots \e_{q_{p-1}}^*e_{q_p} .
%$$ 
  The $S^p$-valued Schatten class $S^p(S^p)$ is the space of those
  operators $x$ from $\ell_2$ to $S^p$ such that
  $\|x\|_p^p=\tr(\tr|x|^p)<\infty$, where the inner trace is the
  $S^p$-valued analogue of the usual trace.  The $S^p$-valued entry
  space $S^p_I(S^p)$ is the
  % space of those $x\in S^p(S^p)$ such that $x_q=\tr\e_q^*x
  closed subspace spanned by the $x_q\e_q$ with $x_q\in S^p$ and $q\in
  I$: $x_q=\tr\e_q^*x$ is the operator coefficient of $x$ at matrix
  coordinate $q$. Thus, for even integers $p$ and $x=(x_q)_{q\in
    I}=\sum_{q\in I}x_q\e_q$ with $x_q\in S^p$ and $I$ finite,
  \begin{equation*}
    \|x\|_p^p=\sum_{q_1,\dots q_p\in I}\tr x_{q_1}^*x_{q_2}\dots
    x_{q_{p-1}}^*x_{q_p}\tr \e_{q_1}^*\e_{q_2}\dots
    \e_{q_{p-1}}^*e_{q_p}.
  \end{equation*}

  A Schur multiplier\index{Schur multiplier} $T$ on $S^p_I$ associated
  to $(\mu_q)_{q\in I}\in\C^I$ is a bounded operator on $S^p_I$ such
  that $T\e_q=\mu_q\e_q$ for $q\in I$.  $T$ is furthermore completely
  bounded\index{completely bounded Schur multiplier} (c.b.~for short)
  if $T$ is bounded as the operator on $S^p_I(S^p)$ defined by
  $T(x_q\e_q)=\mu_qx_q\e_q$ for $x_q\in S^p$ and $q\in I$.

  We shall stick to this harmonic analysis type notation; let us
  nevertheless show how these objects are termed with tensor products:
  $S^p(S^p)$ is also $S^p(\ell_2\otimes_2\ell_2)$ endowed with
  $\|x\|_p^p=\tr\otimes\tr|x|^p$; one should write
  $x_q\otimes\e_q$ instead of $x_q\e_q$; here
  $x_q=\Id_{S^p}\otimes\tr((\Id_{\ell_2}\otimes\e_q^*)x)$; $T$ is
  c.b.~if $\Id_{S^p}\otimes T$ is bounded on
  $S^p(\ell_2\otimes_2\ell_2)$.
\end{Not}

\begin{acknowledgements}
  The first-named and last-named authors undertook this research at
  the \'Equipe d'Analyse de l'Universit\'e Paris 6. It is their
  pleasure to acknowledge its kind hospitality.
\end{acknowledgements}

%
%
%
%
\section{Definitions}
%
%
%
%
%
%
%
\noindent We use the notion of unconditionality in order to define the matrix
analogue of Rudin's ``commutative'' $\Lambda(p)$ sets.
%
\begin{dfn}
%
  Let $X$ be a Banach space. The sequence $(y_n)\subseteq X$ is an
  unconditional basic sequence in $X$ if there is a constant $D$ such
  that
  \begin{equation*}
    \Bigl\|\sum\vartheta_nc_ny_n\Bigr\|_X\le D\Bigl\|\sum
    c_ny_n\Bigr\|_X
  \end{equation*} 
  for every real \klammer{vs.~complex} choice of signs
  $\vartheta_n\in\{-1,1\}$ \klammer{vs.~$\vartheta_n\in\T$} and every
  finitely supported sequence of scalar coefficients $(c_n)$. The
  optimal $D$ is the real \klammer{vs.~complex} unconditionality
  constant of $(y_n)$ in $X$.
%
\end{dfn}
%
Real and complex unconditionality are isomorphically equivalent: the
complex unconditionality constant is at most $\pi/2$ times the real
one. The notions of unconditionality and multipliers are intimately
connected: we have
%
\begin{prp}
%
  Let $(y_n)\subseteq X$ be an unconditional basic sequence in $X$ and
  let $Y$ be the closed subspace of $X$ spanned by $(y_n)$. The real
  \klammer{vs.~complex} unconditionality constant of $(y_n)$ in $X$ is
  exactly the least upper bound for the norms
  $\|T\|_{\mathscr{L}(Y)}$, where $T$ is the multiplication operator
  defined by $Ty_n=\mu_ny_n$, and the $\mu_n$ range over all real
  \klammer{vs.~complex} numbers with $|\mu_n|\le1$.
  % and $\vartheta$ ranges over all choices of signs
  % $\vartheta_n\in\{-1,1\}$ \klammer{vs.~$\vartheta_n\in\T$}.
%
\end{prp}
%
Let us encompass the notions proposed in Question $(\dagger)$.

\begin{dfn}
  Let $I\subseteq \Row\times \Col $ and $p>2$.
  {\renewcommand{\labelenumi}{$(\alph{enumi})$}
    \begin{enumerate}
    \item \cite[Def.~4.1]{ha98} $I$ is a $\sigma(p)$ set if
  $(\e_q)_{q\in I}$ is an unconditional basic sequence in $S^p$. This
  amounts to the uniform boundedness of the family of all relative
  Schur multipliers by signs
%
  \begin{equation}\label{sigma:schur}
%
    T_\vartheta\colon S^p_I\to S^p_I,~x=(x_q)_{q\in I}\mapsto
    T_\vartheta x=(\vartheta_qx_q)_{q\in I}\text{ with }
    \vartheta_q\in\{-1,1\}.
%
  \end{equation}
%
  By \cite[Lemma 0.5]{ha98}, this means that there is a constant $D$
  such that for every finitely supported operator $x=(x_q)_{q\in
    I}=\sum_{q\in I}x_q\e_q$ with $x_q\in\C$
  \begin{equation}\label{def:1}
    D^{-1}\|x\|_p\le|\!|\!|x|\!|\!|_p\le\|x\|_p,
  \end{equation}
  where the second inequality is a convexity inequality that is always
  satisfied \klammer{see \cite[Theorem~8.9]{si80}} and
  \begin{equation}\label{def:2}
    |\!|\!|x|\!|\!|_p^p=
    \sum_c\Bigl(\sum_r|x_{rc}|^2\Bigr)^{p/2}
    \vee
    \sum_r\Bigl(\sum_c|x_{rc}|^2\Bigr)^{p/2}.
  \end{equation}
\item \cite[Def.~4.4]{ha98} $I$ is a complete $\sigma(p)$ set if the
  family of all relative Schur multipliers by signs
  \eqref{sigma:schur} is uniformly c.b. By \cite[Lemma 0.5]{ha98}, $I$
  is completely $\sigma(p)$ if and only if there is a constant $D$
  such that for every finitely supported operator valued operator
  $x=(x_q)_{q\in I}=\sum_{q\in I}x_q\e_q$ with $x_q\in S^p$
  \begin{equation}\label{knc}
    D^{-1}\|x\|_p\le|\!|\!|x|\!|\!|_p\le\|x\|_p,
  \end{equation}
  where the second inequality is a convexity inequality that is always
  satisfied and
  \begin{equation*}
    |\!|\!|x|\!|\!|_p^p=
    \sum_c
    \Bigl\|\Bigl(\sum_rx_{rc}^*x_{rc}\Bigr)^{1/2}\Bigr\|_p^p\vee
    \sum_r
    \Bigl\|\Bigl(\sum_cx_{rc}x_{rc}^*\Bigr)^{1/2}\Bigr\|_p^p.
  \end{equation*}
\end{enumerate}}
\end{dfn}
%
The notion of a complete $\sigma(p)$ set is stronger than that of a
$\sigma(p)$ set: Inequality \eqref{def:1} amounts to Inequality
\eqref{knc} tested on operators of the type $x=\sum_{q\in I}x_q\e_q$
with each $x_q$ acting on the same one-dimensional subspace of
$\ell_2$. It is an important open problem to decide whether the
notions differ. An affirmative answer would solve Pisier's conjecture
about completely bounded Schur multipliers \cite[p.~113]{pi01}.
% the Hilbert space $\C$.%=x'_q\e_{00}$, $x'_q\in\C$.


Notorious examples of 1-unconditional basic sequences in all
Schatten classes $S^p$ are single co\-lumns, single rows, single
diagonals and single anti-diagonals --- and more generally ``column sets''
(vs.~``row sets'') $I$ such that for each $(r,c)\in I$, no other
element of $I$ is in the column $c$ (vs.~row $r$). These sets are
called sections in \cite[Def.~4.3]{va69}
% , $(\beta)$ single diagonals and single anti-diagonals and more
% generally sets $I$ such that for each $(r,c)\in I$, there is no
% other element of $I$ in the row $r$ nor in the column $c$.

We shall try to express these notions in terms of trails on bipartite
graphs.  We proceed as announced in the Introduction: then each
example above is a union of disjoint star graphs in which one vertex
of one class is connected to some vertices of the other class: trails
in a star graph have at most length 2.
% A graph $(\beta)$ contains only trails of length at most
% 1. $\Row\times \Col$ is the complete bipartite graph on
% $V=\Row\amalg \Col$.
%
\begin{dfn}\label{def2}
%
  Let $I\subseteq \Row\times \Col $ and $s\ge1$ an integer. We
  consider $I$ as a bipartite graph: its vertex set is $V=\Row\amalg
  \Col$ and its edge set is $E=\bigl\{\,\{r,c\}\subseteq V:(r,c)\in
  I\,\bigr\}$.
  % edges join only column vertices $c\in \Col$ with row vertices
  % $r\in \Row$, this occurring exactly if $(r,c)\in I$.

  $(a)$ The sets of trails of length $s$ on the graph $I$ from the
  column \klammer{vs.~row} vertex $v_0$ to the vertex $v_s$ are
  respectively
  \begin{equation*}
    \mathscr{\Col}^s(I;v_0,v_s)=\{\,(v_0,\dots,v_s)\in V^{s+1}: v_0\in
    \Col~\text{\& all}~\{v_i,v_{i+1}\}\in E~\text{are distinct}\,\},
  \end{equation*}
  \begin{equation*}
    \mathscr{\Row}^s(I;v_0,v_s)=\{\,(v_0,\dots,v_s)\in V^{s+1}: v_0\in
    \Row~\text{\& all}~\{v_i,v_{i+1}\}\in E~\text{are distinct}\,\}.
  \end{equation*}

  $(b)$ We define the Rudin numbers of trails starting respectively
  with a column vertex and a row vertex by
  $c_s(I;v_0,v_s)=\#\mathscr{\Col}^s(I;v_0,v_s)$ and
  $r_s(I;v_0,v_s)=\#\mathscr{\Row}^s(I;v_0,v_s)$.
%
\end{dfn}

\begin{rem}
  In other words, for an integer $l\ge1$,
  \begin{eqnarray*}
    c_{2l-1}(I;v_0,v_{2l-1})&=&\#\left[\vcenter{
        \hbox{$(r_1,c_1),(r_1,c_2),(r_2,c_2),(r_2,c_3),\dots,(r_l,c_l)$}
        \hbox{pairwise distinct in $I:c_1=v_0,r_l=v_{2l-1}$}}\right]\\
    c_{2l}(I;v_0,v_{2l})&=&\#\left[\vcenter{
        \hbox{$(r_1,c_1),(r_1,c_2),\dots,(r_l,c_l),(r_l,c_{l+1})$}
        \hbox{pairwise distinct in $I:c_1=v_0,c_{l+1}=v_{2l}$}}\right]
  \end{eqnarray*}
  and similarly for $r_s(I;v_0,v_s)$.  If $s$ is odd, then
  $c_s(I;v_0,v_s)=r_s(I;v_s,v_0)$ for all $(v_0,v_s)\in \Col\times
  \Row$. But if $s$ is even, one Rudin number may be bounded while the
  other is infinite: see \cite[Rem.~6.4$(ii)$]{ha98}.
\end{rem}
%
%
%
%
%
\section{\texorpdfstring{$\sigma(p)$ sets as matrix $\Lambda(p)$ sets}{\textsigma(p) sets as matrix \textLambda(p) sets}}
%
%
%
%
%
%
%
%
\noindent We claim the following result.
%
\begin{thm}\label{MAIN}
%
  Let $I\subseteq \Row\times \Col $ and $p=2s$ be an even integer.  If
  $I$ is a union of sets $I_1,\dots,I_l$ such that one of the Rudin
  numbers $c_s(I_j;v_0,v_s)$ or $r_s(I_j;v_0,v_s)$ is a bounded
  function of $(v_0,v_s)$, for each $j$, then $I$ is a complete
  $\sigma(p)$ set.
%
\end{thm}
%
This follows from Theorem \ref{sig:thm} below: the union of two
complete $\sigma(p)$ sets is a complete $\sigma(p)$ set by
\cite[Rem.~after Def.~4.4]{ha98}; furthermore the transposed set
$I^*=\{\,q^*:q\in I\,\}\subseteq \Col\times \Row$ is a complete
$\sigma(p)$ set provided $I$ is.  Note that the case of
$\sigma(\infty)$ sets (see \cite[Rem.\ 4.6$(iii)$]{ha98}) provides
evidence that Theorem \ref{MAIN} might be a characterisation of
complete $\sigma(p)$ sets for even $p$.
%
\begin{thm}\label{sig:thm}
%
  Let $I\subseteq \Row\times \Col $ and $p=2s$ be an even integer. If
  the Rudin number $c_s(I;v_0,v_s)$ is a bounded function of
  $(v_0,v_s)$, then $I$ is a complete $\sigma(p)$ set.
%
\end{thm}
%
This is proved for $p=4$ in \cite[Prop.~6.5]{ha98}. We wish to
emphasise that the proof below follows the scheme of the proof of
\cite[Theorem~1.13]{ha98}. In particular, we make crucial use of
Pisier's idea to express repetitions by dependent Rademacher variables
(\cite[Prop.~1.14]{ha98}).

\begin{proof}
  Let $x=\sum_{q\in I}x_q\e_q$ with $x_q\in S^p$. We have the
  following expression for $\|x\|_p$.
  \begin{equation*}
    \|x\|_p^p=\tr\otimes\tr (x^*x)^s=\|y\|_2^2\quad
    \text{with}\quad y=\overbrace{x^*xx^*\cdots x\pstar}^{s~\text{terms}},
  \end{equation*}
  i.e., $y$ is the product of $s$ terms which are alternatively $x^*$
  and $x$, and we set $x\pstar=x$ for even $s$, $x\pstar=x^*$ for odd
  $s$.
  % We shall work on $y$ and suppose that $c_s(I;v_0,v_s)$ is bounded.
  % If we suppose on the other hand that $r_s(I;v_0,v_s)$ is bounded,
  % we would have to consider $y^*$ instead of $y$ in the sequel.
  Set $\Col\pstar=\Col$ for even $s$ and $\Col\pstar=\Row$ for odd
  $s$.  Let $(v_0,v_s)\in \Col\times \Col\pstar$ and
  $y_{v_0v_s}=\tr\e_{v_0v_s}^*y$ be the matrix coefficient of $y$
  at coordinate $(v_0,v_s)$. Then we obtain by the rule of matrix
  multiplication
%
  \begin{gather}
%
    y=\sum_{q_1,\dots,q_s\in I}
    (x_{q_1}^*\e_{q_1}^*)(x_{q_2}\e_{q_2})\dots
    (x_{q_s}\pstar\e_{q_s}\pstar)\notag\\
    \label{yvw}
    y_{v_0v_s}=\sum_{
      \scriptstyle (v_1,v_0),\,(v_1,v_2),\dots\,\in I
      % v\in\mathscr{\Col}^s(I;v_0,v_s)
    }
    x_{v_1v_0}^*x_{v_1v_2}x_{v_3v_2}^*\dots x_{( v_{s-1},v_s
      )\ppstar}\pstar.
%
  \end{gather}
  Let $\mathscr{E}$ be the set of equivalence relations on
  $\{1,\dots,s\}$. Then
  % , by the expression of the Hilbert--Schmidt norm and by the
  % Arithmetic-Quadratic Mean Inequality,
  \begin{equation}\label{sim}
    % \|x\|_p^p=\sum_{( v_0,v_s)\in \Col\times \Col\ppstar}
    % \|y_{v_0v_s}\|_2^2\le \#\mathscr{E}
    y=\sum_{\sim\in\mathscr{E}}
    % \Bigl\|
    \sum_{i\sim j\Leftrightarrow q_i=q_j}
    (x_{q_1}^*\e_{q_1}^*)(x_{q_2}\e_{q_2})\dots
    (x_{q_s}\pstar\e_{q_s}\pstar).
    % \Bigr\|_2^2
  \end{equation}
  We shall bound the sum above in two steps.

  $(a)$ Let $\sim$ be equality and consider the corresponding term in
  the sum \eqref{sim}. The number of terms in the sum \eqref{yvw} such
  that $\{v_{i-1},v_i\}\ne\{v_{j-1},v_j\}$ if $i\ne j$ is
  $c_s(I;v_0,v_s)$.  If $c$ is an upper bound for $c_s(I;v_0,v_s)$, we
  have by the expression of the Hilbert--Schmidt norm and the
  Arithmetic-Quadratic Mean Inequality
%
  \begin{eqnarray*}
%
    \lefteqn{\Bigl\|
      \sum_{\substack{
          q_1,\dots,q_s\\
          \text{pairwise distinct}}}
      (x_{q_1}^*\e_{q_1}^*)(x_{q_2}\e_{q_2})\dots
      (x_{q_s}\pstar\e_{q_s}\pstar)
      \Bigr\|_2^2}\qquad\qquad&&\\
    &=&\sum_ {( v_0,v_s)\in \Col\times \Col\ppstar}
    \Bigl\|
    \sum_{\scriptstyle v\in\mathscr{\Col}^s(I;v_0,v_s)
      % \atop\scriptstyle v_i\text{\scriptsize\ distinct}
    }x_{v_1v_0}^*x_{v_1v_2}x_{v_3v_2}^*\dots
    x_{(v_{s-1},v_s)\ppstar}\pstar
    \Bigr\|_2^2
    \\
    &\le&c\sum_ {( v_0,v_s)\in \Col\times \Col\ppstar}
    \sum_{\scriptstyle v\in\mathscr{\Col}^s(I;v_0,v_s)
      % \atop\scriptstyle v_i\text{\scriptsize\ distinct}
    }\Bigl\| x_{v_1v_0}^*x_{v_1v_2}x_{v_3v_2}^*\dots
    x_{(v_{s-1},v_s)\ppstar}\pstar
    \Bigr\|_2^2
    \\
    &=&c\sum_{\substack{
        q_1,\dots,q_s\\
        \text{pairwise distinct}}}
    \bigl\|(x_{q_1}^*\e_{q_1}^*)(x_{q_2}\e_{q_2})\dots(x_{q_s}\pstar\e_{q_s}\pstar)
    \bigr\|_2^2\\
    &\le&c\sum_{q_1,\dots,q_s}
    \bigl\|(x_{q_1}^*\e_{q_1}^*)(x_{q_2}\e_{q_2})\dots(x_{q_s}\pstar\e_{q_s}\pstar)
    \bigr\|_2^2\\
    &=&c\Bigl\|\sum_{q_1,\dots,q_s}
    |(x_{q_1}^*\e_{q_1}^*)(x_{q_2}\e_{q_2})\dots
    (x_{q_s}\pstar\e_{q_s}\pstar)|^2
    \Bigr\|_1
%
  \end{eqnarray*}
  Now this last expression may be bounded accordingly to
  \cite[Cor.~0.9]{ha98} by
  \begin{equation}\label{0.5}
    c\Bigl(\Bigl\|\sum(x_q^*\e_q^*)(x_q\e_q)\Bigr\|_s\vee \Bigl\|\sum
    (x_q\e_q)(x_q^*\e_q^*)\Bigr\|_s\Bigr)^s =c|\!|\!|x|\!|\!|_p^p:
  \end{equation}
  see \cite[Lemma 0.5]{ha98} for the last equality.

  $(b)$ Let $\sim$ be distinct from equality. The corresponding term
  in the sum \eqref{sim} cannot be bounded directly. Consider instead
  \begin{equation*}
    \Psi(\sim)= \Bigl\| \sum_{i\sim j\Rightarrow q_i=q_j}
    (x_{q_1}^*\e_{q_1}^*)(x_{q_2}\e_{q_2})\dots
    (x_{q_s}\pstar\e_{q_s}\pstar) \Bigr\|_2 =\Bigl\| \sum_{i\sim j\Rightarrow
      q_i=q_j} \prod_{i=1}^sf_i(q_i)\Bigr\|_2
  \end{equation*} 
  with $f_i(q)=x_q\e_q$ for even $i$ and $f_i(q)=x_q^*\e_q^*$ for odd
  $i$. We may now apply Pisier's Lemma \cite[Prop.~1.14]{ha98}: let
  $0\le r\le s-2$ be the number of one element equivalence classes
  modulo $\sim$; then
%
  \begin{equation}\label{PisHar}
%
    \Psi(\sim)\le \|x\|_p^r(B|\!|\!|x|\!|\!|_p)^{s-r},
%
  \end{equation}
%
  where $B$ is the constant arising in Lust-Piquard's non-commutative
  Khinchin inequality. In order to finish the proof, one does an
  induction on the number of atoms of the partition induced by $\sim$,
  along the lines of step 2 of the proof of \cite[Theorem~1.13]{ha98}.
\end{proof}

The Moebius inversion formula for partitions enabled Pisier
\cite{pi99} to obtain the following explicit bounds in the computation
above:
\begin{gather}
  \|y\|_2\le c^{1/2}|\!|\!|x|\!|\!|_p^s+
  \sum_{0\le r\le s-2}\binom sr(s-r)!
  \|x\|_p^r\bigl((3\pi/4)|\!|\!|x|\!|\!|_p\bigr)^{s-r}\notag\\
\label{p-ortho}
  \|x\|_p\le\bigl((4c)^{1/p}\vee 9\pi p/8\bigr)|\!|\!|x|\!|\!|_p.
\end{gather}
Let us also record the following consequence of his study of
$p$-orthogonal sums. The family $(x_q\e_q)_{q\in I}$ is $p$-orthogonal
in the sense of \cite{pi99} if and only if the graph associated to $I$
does not contain any circuit of length $p$, so that we have by
\cite[Theorem~3.1]{pi99}:
%
\begin{thm}\label{piscycle}
%
  Let $p\ge4$ be an even integer. If $I$ does not contain any circuit
  of length $p$, then $I$ is a complete $\sigma(p)$ set with constant
  at most $3\pi p/2$.
%
\end{thm}
%
% \rem\label{rem-pisier}The method of permits to optimise step $(b)$
% and to give the following bound for the $\sigma(p)$ set constant
% $\Col$ of \eqref{def:1} in terms of the Rudin number $c$:
%$$
%\|x\|_p\le(4c)^{1/p}+9\pi p/8.
%$$
\begin{rem}
  Pisier proposed to us the following argument to deduce a weaker
  version of Theorem~\ref{sig:thm} from \cite[Theorem~1.13]{ha98}. Let
  $\Gamma=\T^V$ and $z_v$ denote the $v$th coordinate function on
  $\Gamma$. Associate to $I$ the set $\Lambda=\{\,z_rz_c:(r,c)\in I\,\}$.
  Let still $p=2s$ be an even integer. Then $I$ is a complete
  $\sigma(p)$ set if $\Lambda$ is a complete $\Lambda(p)$ set as
  defined in \cite[Def.~1.5]{ha98}, which in turn holds if $\Lambda$
  has property $Z(s)$ as given in \cite[Def.~1.11]{ha98}. It turns out
  that this condition implies the uniform boundedness of
  \begin{equation*}
    c_t(I;v_0,v_t)\vee r_t(I;v_0,v_t)\quad\text{for $t\le s$, $v_0,v_t\in V$.}
    \end{equation*}
    % and istrictly stronger f
    For $p\ge8$, this implication is strict: in fact, the countable
    union of disjoint cycles of length 4 (``quadrilaterals'')
    \begin{equation*}
      I=\bigcup\nolimits_{i\ge0}
      \bigl\{(2i,2i),(2i,2i+1),(2i+1,2i+1),(2i+1,2i)\bigr\}
    \end{equation*} 
    satisfies $c_t(I;v_0,v_t)\vee r_t(I;v_0,v_t)\le2$ whereas
    $\Lambda$ does not satisfy $Z(s)$ for any $s\ge4$.
  \end{rem}


  \begin{rem}
    \label{remschur}
    Theorem~\ref{MAIN} is especially useful to construct c.b.~Schur
    multipliers: by \cite[Rem.~4.6$(ii)$]{ha98}, if $I$ is a complete
    $\sigma(p)$ set, there is a constant $D$ (the constant $D$ in
    \eqref{knc}) such that for every sequence $(\mu_q)\in\C^{
      \Row\times \Col }$ supported by $I$ and every operator
    $T_\mu:(x_q)\mapsto(\mu_qx_q)$ we have
    \begin{equation*}
      \|T_\mu\|_{\mathscr{L}(S^p(S^p))}\le D\sup_{q\in I}|\mu_q|.  
    \end{equation*}
  \end{rem}
%
%
%
%
%
%
%
%
  \section{\texorpdfstring{The intersection of a $\sigma(p)$ set with a finite product
    set}{The intersection of a \textsigma(p) set with a finite product
    set}}
  % {A necessary bound on the size of $I$}
%
%
%
%
%
%
%
%
  \noindent Let $I\subseteq \Row\times \Col $ considered as a bipartite graph as
  in the Introduction and let $I'\subseteq I$ be the subgraph induced
  by the vertex set $\Col'\amalg \Row'$, with $\Col'\subseteq \Col$ a
  set of $m$ column vertices and $\Row'\subseteq \Row$ a set of $n$
  row vertices. In other words, $I'=I\cap \Row'\times \Col'$.  Let
  $d(v)$ be the degree of the vertex $v\in \Col'\amalg \Row'$ in $I'$:
  in other words,
  \begin{gather*}
    \forall c\in \Col'\quad d(c)=\#[I'\cap \Row'\times\{c\}],\\
    \forall r\in \Row'\quad d(r)=\#[I'\cap\{r\}\times \Col'].
  \end{gather*}
  Let us recall that the dual norm of \eqref{def:2} is
  \begin{equation*}
    |\!|\!|x|\!|\!|_{p'}=\inf_{\substack{
        \alpha,\beta\in {S^{p'}}\\
        \alpha+\beta=x}}
    \biggl(\sum_c\Bigl(\sum_r|\alpha_{rc}|^2\Bigr)^{p'/2}\biggr)^{1/p'}+
    \biggl(\sum_r\Bigl(\sum_c|\beta _{rc}|^2\Bigr)^{p'/2}\biggr)^{1/p'},
  \end{equation*} 
  where $p\ge2$ and $1/p+1/p'=1$ (see \cite[Rem.~after Lemma
  0.5]{ha98}).
%
  \begin{lem}\label{size:lem}
%
    Let
    % $I'\subseteq \Row\times \Col$,
    $1\le p'\le2$ and $x=\sum_{q\in I'}x_q$. Then
    \begin{equation*}
      |\!|\!|x|\!|\!|_{p'}^{p'}
      \ge\doublesum_{(r,c)\in I'}\Bigl(\max\bigl(d(c),d(r)\bigr)^{1/2-1/p'}
      |x_{rc}|\Bigr)^{p'}.
    \end{equation*}
%
  \end{lem}

\begin{proof}
  By the $p'$-Quadratic Mean Inequality and by Minkowski's Inequality,
  \begin{eqnarray*}
    \lefteqn{
      \biggl(\sum_{c\in \Col'}\Bigl(
      \sum_{(r,c)\in I'}|\alpha_{rc}|^2\Bigr)^{p'/2}\biggr)^{1/p'}+
      \biggl(\sum_{r\in \Row'}\Bigl(\sum_{(r,c)\in I'}|\beta_{rc}|^2\Bigr)^{p'/2}
      \biggr)^{1/p'}}&&\\
    &\ge&
    \Bigl(\sum_{c\in \Col'}d(c)^{p'/2-1}\sum_{(r,c)\in I'}|\alpha_{rc}|^{p'}
    \Bigr)^{1/p'}+
    \Bigl(\sum_{r\in \Row'}d(r)^{p'/2-1}\sum_{(r,c)\in I'}|\beta _{rc}|^{p'}
    \Bigr)^{1/p'}\\
    &\ge&
    \Bigl(\doublesum_{(r,c)\in I'}
    \bigl(d(c)^{1/2-1/p'}|\alpha_{rc}|+d(r)^{1/2-1/p'}|\beta_{rc}|\bigr)^{p'}
    \Bigr)^{1/p'}
  \end{eqnarray*}
  The lemma follows by taking the infimum over all $\alpha,\beta$ with
  $\alpha_q+\beta_q=x_q$ for $q\in I'$ as one can suppose that
  $\alpha_q=\beta_q=0$ if $q\notin I$; note further that
  $1/2-1/p'\le0$.
\end{proof}

\begin{thm}\label{dc}
%
  If $I$ is a $\sigma(p)$ set with constant $D$ as in \eqref{def:1},
  then the size $\#I'$ of any subgraph $I'$ induced by $m$ column
  vertices and $n$ row vertices, in other words the cardinal of any
  subset $I'=I\cap \Row'\times \Col'$ with $\#\Col'=m$ and
  $\#\Row'=n$, satisfies
  \begin{eqnarray}
    \#I'&\le&D^2\bigl(m^{1/p}n^{1/2}+m^{1/2}n^{1/p}\bigr)^2\label{dens}\\
    \nonumber&\le&4D^2\min(m,n)^{2/p}\max(m,n).\end{eqnarray}
  The exponents in this inequality are optimal even for a complete
  $\sigma(p)$ set $I$ in the following cases:

  $(a)$ if $m$ or $n$ is fixed;

  $(b)$ if $p$ is an even integer and $m=n$.
%
\end{thm}
%
Bound \eqref{dens} holds \emph{a fortiori} if $I$ is a complete
$\sigma(p)$ set. Density conditions thus do not so far permit to
distinguish $\sigma(p)$ sets and complete $\sigma(p)$ sets. One may
conjecture that Inequality \eqref{dens} is also optimal for $p$ not an
even integer and $m=n$: this would be a matrix counterpart to
Bourgain's theorem \cite{bo89} on maximal $\Lambda(p)$ sets.

\begin{proof}
  If \eqref{def:1} holds, then %by \cite[Rem.~4.6$(iv)$]{ha98},
  $\|x|_{I'}\|_p\le D|\!|\!|x|\!|\!|_{p}$ for all $x\in S^p$ by Remark
  \ref{remschur} applied to $(\mu_q)$ the indicator function of $I'$,
  and by duality $|\!|\!|x|_{I'}|\!|\!|_{p'}\le D\|x\|_{p'}$ for all
  $x\in S^{p'}$ (compare with \cite[Rem.~4.6$(iv)$]{ha98}). Let
  % $x$ be defined by
%$$
%x_{rc}=\cases{ d(c)^{1/p'-1/2}+d(r)^{1/p'-1/2}&for $(r,c)\in I'$\cr
%  0&otherwise.\cr}
%$$
  \begin{gather*}
    y=\doublesum_{(r,c)\in I'}d(c)^{1/p'-1/2}\e_{rc},\\
    z=\doublesum_{(r,c)\in I'}d(r)^{1/p'-1/2}\e_{rc},
  \end{gather*}
  Then the $n$ rows of $y$ are all equal, as well as the $m$ columns
  of $z$: $y$ and $z$ have rank 1
  % are elementary tensor products
  and a single singular value. By the norm inequality followed by the
  $(2/p'-1)$-Arithmetic Mean Inequality,
  \begin{eqnarray*}
    \|y+z\|_{p'}&\le&\|y\|_{p'}+\|z\|_{p'}\\
    &=&n^{1/2}\Bigl(\sum_{c\in \Col'}d(c)^{2/p'-1}\Bigr)^{1/2}+
    m^{1/2}\Bigl(\sum_{r\in \Row'}d(r)^{2/p'-1}\Bigr)^{1/2}\\
    &\le&n^{1/2}m^{1-1/p'}(\#I')^{1/p'-1/2}+
    m^{1/2}n^{1-1/p'}(\#I')^{1/p'-1/2}.
  \end{eqnarray*}
  We used that $\sum_{c\in \Col'}d(c)=\sum_{r\in \Row'}d(r)=\#I'$. By
  Lemma \ref{size:lem} applied to $x=y+z$,
  \begin{equation*}
    (\#I')^{1/p'}\le
    D(n^{1/2}m^{1-1/p'}+m^{1/2}n^{1-1/p'})(\#I')^{1/p'-1/2}, 
  \end{equation*} 
  and we get therefore the first part of the theorem.

  Let us show optimality in the given cases.

  $(a)$ Suppose that $n$ is fixed and $\Col'=\Col$: $I'=\Row'\times
  \Col$ is a complete $\sigma(p)$ set for any $p$ as a union of $n$
  rows and $\#I'=n\cdot m$.

  $(b)$ is proved in \cite[Theorem~4.8]{ha98}.
\end{proof}

\begin{rem}
  If $n\nsim m$, the method used in \cite[Theorem~4.8]{ha98} does not
  provide optimal $\sigma(p)$ sets but the following lower bound.  Let
  $p=2s$ with $s\ge2$ an integer. Consider a prime $q$ and let
  $k=s^{s-1}q^s$. By \cite[4.7]{ru60} and \cite[Theorem~2.5]{ha98},
  there is a subset $F\subseteq\{0,\dots,k-1\}$ with $q$ elements
  whose complete $\Lambda(2s)$ constant is independent of $q$. Let
  $m\ge k$ and $0\le n\le m$ and consider the Hankel set
  \begin{equation*}
    % $m=(s+1)s^{s-1}q^s-1$ and
    I=\bigl\{\,(r,c)\in\{0,\dots,n-1\}\times\{0,\dots,m-1\}:r+c\in
    F+m-k\,\bigr\}.  
  \end{equation*} 
  Then the complete $\sigma(p)$ constant of $I$ is independent of $q$
  by \cite[Prop.~4.7]{ha98} and
  \begin{equation*}
    \#I\ge
    \begin{cases}
      nq&\text{if $n\le m-k+1$}\\
      % =s^sq^s$\crs^sq^{s+1}\ge(n+1)qs/(s+1)&if $s^sq^s<n\le
      % m$.\cr}$$
      (m-k+1)q&\text{if $n\ge m-k+1$.}
    \end{cases}
  \end{equation*} 
  If we choose $m=(s+1)k-1$, this
  yields \begin{equation*}\#I\ge\frac{s^{1/s}}{(s+1)^{1+1/s}}\min(n,m)\max(m,n)^{1/s}.
  \end{equation*}
  % In each case, we thus obtain sets $\#I\ge
  % d\min(n,m)\max(m,n)^{1/s}$, where $d$ is independent of
  % \ge%$m$ and $n$, and that shows optimality in
  % the case $m=n$.  This construction yields optimal complete
  % $\sigma(p)$ sets if $n=m$.
  Random construction \ref{rc} provides bigger sets than this
  deterministic construction; however, it also does not provide sets
  that would show the optimality of Inequality \eqref{dens} unless
  $s=2$.
\end{rem}









\section{Circuits in graphs}






\noindent Non-commutative methods yield a new proof to a theorem of Erd\H os
\cite[p.~33]{er64}. Note that its generalisation by Bondy and
Simonovits \cite{bs74} is stronger than Theorem~\ref{eee} below as it
deals with cycles instead of circuits. By Theorem~\ref{piscycle} and
\eqref{dens}
%
\begin{thm}\label{eee}
  Let $p\ge 4$ be an even integer. If $G$ is a nonempty graph on $v$
  vertices with $e$ edges without circuit of length $p$, then
  \begin{equation*}e\le18\pi^2 p^2\,v^{1+2/p}.\end{equation*} If $G$
  is furthermore a bipartite graph whose two vertex classes have
  respectively $m$ and $n$ elements, then
%
  \begin{eqnarray}\label{luc}
    e\le9\pi^2p^2\min(m,n)^{2/p}\max(m,n).
  \end{eqnarray}
%
\end{thm}
% 
\begin{proof}
  For the first assertion, recall that a graph $G$ with $e$ edges
  contains a bipartite subgraph with more than $e/2$ edges (see
  \cite[p.~xvii]{bo78}).
\end{proof}

\begin{rem}
  \L uczak showed to us that \eqref{luc} cannot be optimal if $m$ and
  $n$ are of very different order of magnitude. In particular, let $p$
  be a multiple of 4. Let $e'$ be the maximal number of edges of a
  graph on $n$ vertices without circuit of length $p/2$. If $m>pe'$,
  he shows that \eqref{luc} may be replaced by $e<3m$.
\end{rem}
 
We also get the following result, which enables us to conjecture a
generalisation of the theorems of Erd\H{o}s and Bondy and Simonovits.
%
\begin{thm}\label{ggg}
  Let $G$ be a nonempty graph on $v$ vertices with $e$ edges. Let
  $s\ge 2$ be an integer.

  $(i)$ If \begin{equation*}e>
    % {8sD^2\over (s+1)^{1+1/s}}
    8D^2\,v^{1+1/s}\quad\text{with }D>9\pi s/4,\end{equation*} then
  % $G$ contains a circuit of length $4\le l\le2s$. In fact,
  one may choose two vertices $v_0$ and $v_s$ such that $G$ contains
  more than $D^{2s}/4$ pairwise distinct trails from $v_0$ to $v_s$,
  each of length $s$ and with pairwise distinct edges.

  $(ii)$ One may draw the same conclusion if $G$ is a bipartite graph
  whose two vertex classes have respectively $m$ and $n$ elements and
  \begin{equation*}e>4D^2\min(m,n)^{1/s}\max(m,n)\quad\text{with
    }D>9\pi s/4.\end{equation*}
  % then $G$ contains a circuit of length $4\le l\le2s$. In fact, one
  % may choose two vertices $v_0$ and $v_s$ such that $G$ contains
  % more than $D^{2s}/4$ pairwise distinct trails from $v_0$ to $v_s$,
  % each of length $s$ and with pairwise distinct edges.
\end{thm}
\begin{proof}
  $(i)$ According to \cite[p.~xvii]{bo78}, the graph $G$ contains a
  bipartite subgraph with more than $e/2$ edges,
  % and two vertex classes with $\lfloor v/2\rfloor$ and $\lceil
  % v/2\rceil$ elements each,
  so that we may apply $(ii)$.

  $(ii)$ Combining inequalities \eqref{p-ortho} and \eqref{dens}, if
  $D>9\pi s/4$, then there are vertices $v_0$ and $v_s$ such that the
  number $c$ of pairwise distinct trails from $v_0$ to $v_s$, each of
  length $s$ and with pairwise distinct edges, satisfies
  $(4c)^{1/2s}>D$.
\end{proof}

Two paths with equal endvertices are called independent if they have
only their endvertices in common.

\begin{Q}
  Let $G$ be a graph on $v$ vertices with $e$ edges. Let $s,l\ge 2$ be
  integers. Is it so that there is a constant $D$ such that if
  $e>Dv^{1+1/s}$, then $G$ contains $l$ pairwise independent paths of
  length $s$ with equal endvertices?
\end{Q}

\begin{rem}
  Note that by Theorem~\ref{dc}, the exponent $1+1/s$ is optimal in
  Theorem~\ref{ggg}$(i)$, whereas optimality of the exponent $1+2/p$
  in Theorem~\ref{eee} is an important open question in Graph Theory
  (see \cite{luw99}).
\end{rem}

One may also formulate Theorem~\ref{ggg}$(ii)$ in the following way.
%
\begin{thm}\label{fff}
%
  % For each integer $c$ and each even integer $s$, there is a
  % constant $E$ such that i
  If a bipartite graph $G_2(n,m)$ with $n$ and $m$ vertices in its two
  classes avoids
  % not contain
  any union of $c$ pairwise distinct trails along $s$ pairwise
  distinct edges between two given vertices as a subgraph, where the
  class of the first vertex is fixed, then the size $e$ of the graph
  satisfies
  \begin{equation*}
    e\le 4\max((4c)^{1/2s},9\pi s/4))\min(m,n)^{1/s}\max(m,n).
  \end{equation*}
%
\end{thm}
%
%
%
%
%
%
\section{A random construction of graphs}
%
%
%
%
%
\noindent Let us precise our construction of a random graph.
%
\begin{construction}\label{rc}
  Let $\Col,\Row$ be two sets such that $\#\Col=m$ and $\#\Row=n$. Let
  $0\le\alpha\le1$.  A random bipartite graph on $V=\Col\amalg \Row$
  is defined by selecting independently each edge in
  $E=\bigl\{\,\{r,c\}\subseteq V:(r,c)\in \Row\times \Col\,\bigr\}$ with
  the same probability $\alpha$. The resulting random edge set is
  denoted by $E'\subseteq E$ and $I'\subseteq \Row\times \Col$ denotes
  the associated random subset.
\end{construction}
%
Our aim is to construct large sets while keeping down the Rudin number
$c_s$.
%
\begin{thm}\label{randcon}
  For each $\eps>0$ and for each integer $s\ge2$, there is an $\alpha$
  such that Random construction \ref{rc} yields subsets $I'\subseteq
  \Row\times \Col$ with size
  \begin{equation*}\#I'\sim\min(m,n)^{1/2+1/s}\max(m,n)^{1/2-\eps}\end{equation*}
  and with $\sigma(2s)$ constant independent of $m$ and $n$ for
  $mn\to\infty$.
\end{thm}

\begin{proof}
  Let us suppose without loss of generality that $m\ge n$. We want to
  estimate the Rudin number of trails in $I'$.  Set
  ${\Col}\pstar=\Col$ for even $s$, ${\Col}\pstar=\Row$ for odd $s$
  and let $(v_0,v_s)\in \Col\times{\Col}\pstar$.  Let $l\ge1$ be a
  fixed integer. Then
  \begin{align*}
    \Prob[c_s(I';v_0,v_s)\ge l]&=
      \Prob[\exists~l~\text{distinct trails}~(v_0^j,\dots,v_s^j)~\text{in}~
      \mathscr{\Col}^s(I';v_0,v_s)]\\
    &=\Prob[\,E'\supseteq\bigl\{\{v_{i-1}^j,v_i^j\}\bigr\}_{i,j}:
    \{(v_0^j,\dots,v_s^j)\}_{j=1}^l\subseteq\mathscr{\Col}^s(\Row\times \Col;v_0,v_s)\,]\\
    &\le\sum_{k=\lceil l^{1/s}\rceil}^{ls}\#A_k\cdot\alpha^k,
  \end{align*}
  where $A_k$ is the following set of $l$-element subsets of trails in
  $\mathscr{\Col}^s(\Row\times \Col;v_0,v_s)$ built with $k$ pairwise
  distinct edges 
  \begin{equation*}
    A_k=\Bigl\{\,\{(v_0^j,\dots,v_s^j)\}_{j=1}^l\subseteq\mathscr{\Col}^s(\Row\times
    \Col;v_0,v_s): \#\bigl\{\{v_{i-1}^j,v_i^j\}\bigr\}_{i,j}=k\,\Bigr\};
  \end{equation*} the lower limit of summation is $\lceil l^{1/s}\rceil$
  because one can build at most $k^s$ pairwise distinct trails of length
  $s$ with $k$ pairwise distinct edges.

  In order to estimate $\#A_k$, we now have to bound the number of
  pairwise distinct vertices and the number of pairwise distinct
  column vertices in each set of $l$ trails
  $\{(v_0^j,\dots,v_s^j)\}_{j=1}^l\in A_k$. We claim that
%
  \begin{eqnarray}\label{v1}
    \#\{\,v_i^j:1\le i\le s-1,1\le j\le l\,\}
    &\le&
    % l(s-1)-\lceil(s-1)/s(ls-k)\rceil=\lfloor k(s-1)/s\rfloor
    k(s-1)/s,\\
    \#\{\,v_{2i}^j:1\le i\le\lceil s/2 \rceil -1, 1\le j\le l\,\}
    &\le& k/2.\label{v2}
  \end{eqnarray}
%
  % For each $j\ge2$, let $a_j$ be the number of edges
  % $\{v_{i-1}^j,v_i^j\}$, $1\le i\le s$, that repeat an edge in
  % $\bigl\{\{v_{i-1}^{j'},v_i^{j'}\}:1\le i\le s,1\le
  % j'<j\bigr\}$. If $a_j=s$, then all $s-1$ vertices $v_i^j$, $1\le
  % i\le s-1$, are repetitions, and thus all $\lfloor(s-1)/2\rfloor$
  % column vertices; otherwise at least $a_j$ vertices among $v_i^j$,
  % $1\le i\le s-1$, are repeated, and thus at least $\lfloor
  % a_j/2\rfloor$ column vertices. As $a_2+\dots+a_l=ls-k$, at least
  % $(ls-k)\cdot(s-1)/s$ vertices among $l(s-1)$ are repeated, and
  % this proves the first claim.
  %% If $s$ is odd, the second claim is trivial; If $s$ is even,
  % The number of repeated column vertices among all
  % $l\lfloor(s-1)/2\rfloor$ is at least
  % \begin{eqnarray*}
  %   \sum_{j:a_j=s}\Bigl\lfloor{s-1\over2}\Bigr\rfloor+\sum_{j:a_j\ne s}\Bigl\lfloor{a_j\over2}\Bigr\rfloor&\ge&\#[j:a_j=s]\Bigl(\Bigl\lfloor{s-1\over2}\Bigr\rfloor-{s-1\over2}\Bigr)+{ls-l-k\over2}\\
  %   &\ge&\cases{(ls-l-k)/2-(ls-k)/(2s)&if $s$ is even\cr
  %     (ls-l-k)/2&if $s$ is odd.\cr}
  % \end{eqnarray*}
  % This proves the second claim.
  The second estimate is trivial, because each column vertex
  $v_{2i}^j$ accounts for two distinct edges $\{v_{2i-1}^j,v_{2i}^j\}$
  and $\{v_{2i}^j,v_{2i+1}^j\}$. For the first estimate, note that
  each maximal sequence of $h$ consecutive pairwise distinct vertices
  $(v_{a+1}^j,\dots,v_{a+h}^j)$ accounts for $h+1$
  % =h\cdot (h+1)/h \ge h\cdot s/(s-1)$$
  pairwise distinct
  edges \begin{equation*}\{v_a^j,v_{a+1}^j\},\{v_{a+1}^j,v_{a+2}^j\},\dots,\{v_{a+h}^j,v_{a+h+1}^j\};\end{equation*}
  as $h\le s-1$, $h+1\ge hs\slash(s-1)$.
  % , so that $k-k/s$ pairwise distinct vertices in $\{v_i^j:1\le i\le
  % s-1,1\le j\le l\}$ account for at least $(k-k/s)\cdot s/(s-1)$
  % pairwise distinct edges.
  By \eqref{v1} and \eqref{v2}, \begin{equation*} \#A_k\le
    % S^{l(s-1)}_{\lfloor k(s-1)/s\rfloor}
    % S\cdot\cases{(mn)^{k/2-k/(2s)}&if $s$ is even\cr
    %   m^{k/2}n^{k/2-k/s}&if $s$ is odd.\cr}
    m^{k/2}n^{k/2-k/s}(k-k/s)^{ls-l}\le (ls)^{ls}m^{k/2}n^{k/2-k/s}:
    % m^{\lfloor k(s-1)/s\rfloor},
  \end{equation*}
  % $S^n_d$ is the Stirling number of partitions of a set of $n$
  % elements into $d$ subsets.  $S\le(k-k/s)^{ls-l}$ is a
  % combinatorial constant that is independent of $m$ and $n$ (choose
  % first $k/2$ column vertices and $k/2-k/s$ row vertices; then the
  % number of ways to arrange those vertices into an element of $A_k$
  % is bounded by the number of arrangements with repetition of $ls-l$
  % out of $k-k/s$ elements).
  each element of $A_k$ is obtained by a choice of at most $k-k/s$
  vertices, of which at most $k/2$ are column vertices, and the choice
  of an arrangement with repetitions of $ls-l$ out of at most $k-k/s$
  vertices.

  Put
  % $\alpha=(mn)^{-1/2+1/(2s)-\eps}$ if $s$ is even and
  $\alpha=m^{-1/2}n^{-1/2+1/s}(\#\Col\cdot\#\Col\pstar)^{-\eps}$. Then
  \begin{eqnarray*}
    \Prob[\sup_{(v_0,v_s)
      % \in \Col\times{\Col}\ppstar
    }c_s(I';v_0,v_s)\ge l]&\le&
    \#\Col\cdot\#\Col\pstar\cdot(ls)^{ls}\sum_{k=\lceil
      l^{1/s}\rceil}^{ls}(\#\Col\cdot\#\Col\pstar)^{-k\eps}\\
    % S^{l(s-1)}_{\lfloor k(s-1)/s\rfloor} and since, given $n$,
    % $S^n_d$
    % is maximal for $d=\lfloor n/2\rfloor$,
    &\le& (ls)^{ls}
    % S^{l(s-1)}_{\lfloor k(s-1)/s\rfloor} S^{l(s-1)}_{\lfloor
    %   l(s-1)/2\rfloor}
    \frac{(\#\Col\cdot\#\Col\pstar)^{1-\lceil
        l^{1/s}\rceil\eps}}{1-(\#\Col\cdot\#\Col\pstar)^{-\eps}}.
  \end{eqnarray*}
  % For $l$ such that $\lceil l^{1/s}\rceil\eps>2$, this shows that
  % for any positive $\eps$ and for large $m$ and $n$, there are sets
  % with density $(mn)^{1/2+1/(2s)-\eps}$ if $s$ is even and
  % $m^{1/2-\eps}n^{1/2+1/s-\eps}$ if $s$ is odd.  The theorem holds
  % by the Law of Large Numbers.
  Choose $l$ such that $\lceil l^{1/s}\rceil\eps>1$. Then this
  probability is little for $mn$ large. On the other hand, $\#I'$ is
  of order $mn\alpha$ with probability close to 1.
\end{proof}


\begin{rem}
  This construction yields much better results for $s=2$. Keeping the
  notation of the proof above and $m\ge n$, we get $k=2l$, $A_k=\binom
  nl$ and \begin{equation*} \Prob[\sup_{(v_0,v_2)\in \Col\times
      \Col}c_2(I';v_0,v_2)\ge l]\le m^2\binom
    nl\alpha^{2l}.\end{equation*} Let $l\ge2$ and
  $\alpha=m^{-1/l}n^{-1/2}$. This yields sets $I'\subseteq \Row\times
  \Col$ with size \begin{equation*} \#I'\sim n^{1/2}m^{1-1/l}
  \end{equation*} and with $\sigma(4)$ constant independent of $m$ and $n$.
  This case has been extensively studied in Graph theory as the
  ``Zarankiewicz problem:'' if $c_2(I';v_0,v_2)\le l$ for all
  $v_0,v_2\in\Col$, then the graph $I'$ does not contain a complete
  bipartite subgraph on any two column vertices $v_0,v_2$ and $l+1$ row
  vertices. Reiman (see \cite[Theorem~VI.2.6]{bo78}) showed that then
  \begin{equation*}\#I'\le\bigl(lnm(m-1)+n^2/4\bigr)^{1/2}+n/2 \sim
    % _{m\ge n\atop m\to\infty}
    l^{1/2}n^{1/2}m.\end{equation*} With use of finite projective geometries, he also
  showed that this bound is optimal for
  \begin{equation*}n=l\frac{q^{r+1}-1}{q^2-1}\frac{q^r-1}{q-1}
    \quad,\quad m=\frac{q^{r+1}-1}{q-1}\end{equation*} with $q$ a
  prime power and $r\ge2$ an integer, and thus with $m\le n$: there
  seems to be no constructive example of extremal graphs with
  $c_2(I';v_0,v_2)\le l$ and $m>n$ besides the trivial case of
  complete bipartite graphs with $m>n=l-1$.
\end{rem}

\begin{rem}
  In the case $s=3$, our result cannot be improved just by refining
  the estimation of $\#A_k$. If we consider first $l$ distinct paths
  that have their second vertex in common and then $l$ independent
  paths, we get
  \begin{equation*}\#A_{2l+1}\ge\binom mln\quad,\quad
    \#A_{3l}\ge\binom ml\binom nl.\end{equation*} Therefore any choice
  of $\alpha$ as a monomial $m^{-t}n^{-u}$ in the proof above must
  satisfy $t\ge(l+1)\slash(2l+1)$, $t+u\ge(2l+2)\slash(3l)$ and this yields sets
  with
  \begin{equation*}\#I'\preccurlyeq
    m^{1/2-1/2(4l+2)}n^{5/6-(7l+6)/(12l^2+6l)}.\end{equation*}
\end{rem}
% \quest If $n\nsim m$ and $p\ge6$ , we have two cand
%
%
%


% {\small \halign{ #\hfill\quad&#\hfill\quad&#\hfill\quad\cr Asma
%     Harcharras&Stefan Neuwirth&Krzysztof Oleszkiewicz\cr Mathematics
%     Department&Laboratoire de Math\'ematiques&Institute of
%     Mathematics\cr University of Missouri-Columbia&Universit\'e de
%     Franche-Comt\'e&Warsaw University\cr Co\-lum\-bia MO
%     65211&25030~Besan\c con cedex&Banacha 2, 02-097 Warszawa\cr
%     U.S.A.&France&Poland\cr
%     harchars@math.missouri.edu&neuwirth@math.univ-fcomte.fr&koles@mimuw.edu.pl\cr
%   }}



%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "~/h"
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\clearpage

\chapter{Cycles and 1-un\-con\-di\-tio\-nal matrices}
\label{cycl}
\begin{abstract}
  \noindent We characterise the 1-unconditional subsets
  $(\mathrm{e}_{rc})_{(r,c) \in I}$ of the set of elementary matrices
  in the Schatten-von-Neumann class $\mathrm{S}^p$. The set of couples
  $I$ must be the set of edges of a bipartite graph without cycles of
  even length $4 \le p$ if $p$ is an even integer, and without
  cycles at all if $p$ is a positive real number that is not an even
  integer. In the latter case, $I$ is even a Varopoulos set of
  V-interpolation of constant 1. We also study the metric
  unconditional approximation property for the space $\mathrm{S}^p_I$
  spanned by $(\mathrm{e}_{rc})_{(r,c) \in I}$ in $\mathrm{S}^p$.
\end{abstract}
\section{Introduction}

\noindent The starting point for this investigation has been the following
isometric question on the Schatten-von-Neumann class $\Sch^p$.

\begin{Q}\label{q1}
  Which matrix coefficients of an operator $x\in \Sch^p$ must vanish so
  that the norm of $x$ does not depend on the argument, or on the
  sign, of the remaining nonzero matrix coefficients?
\end{Q}

Let $\Col$ be the set of columns and $\Row$ be the set of rows for
coordinates in the matrix. Let $I\subseteq\Row\times\Col$ be the set of
matrix coordinates of the nonzero matrix coefficients of $x$ (the \emph{pattern}.)
Question~\ref{q1} describes the notion of a complex, or real,
1-un\-con\-di\-tio\-nal basic sequence $(\e_{rc})_{(r,c)\in I}$ of
elementary matrices in $\Sch^p$ (see Definition \ref{unc:def}.)

By a convexity argument, Question~\ref{q1} is
equivalent to the following question on Schur multiplication.

\begin{Q}\label{q2}
  Which matrix coefficients of an operator $x\in \Sch^p$ must vanish so
  that for all matrices $\varphi$ of complex, or real, numbers
  \begin{equation*}
    \|\varphi*x\|\le\sup|\varphi_{rc}|~\|x\|,
  \end{equation*}
  where $\varphi*x$ is
  the Schur (or Hadamard or entrywise) product defined by
  \begin{equation*}
    (\varphi*x)_{rc}=\varphi_{rc}x_{rc}?
  \end{equation*}
\end{Q}

In the case $p=\infty$, Grothendieck's inequality yields an estimation
for the norm of Schur multiplication by $\varphi$ in terms of the
projective tensor product $\ell_\Col^\infty\projtens\ell_\Row^\infty$:
this norm is equivalent to the supremum of the norm of those elements
of $\ell_\Col^\infty\projtens\ell_\Row^\infty$ whose
coefficient matrices are finite submatrices of $\varphi$. In the
framework of tensor algebras over discrete spaces, Question~\ref{q2}
turns out to describe as well the isometric counterpart to Varopoulos'
$\Varopoulos$-Sidon sets as well as to his sets of $\Varopoulos$-interpolation. The
following isometric question has however a different answer.

\begin{Q}\label{q3}
  Which coefficients of a tensor
  $u\in\ell_\Col^\infty\projtens\ell_\Row^\infty$ must vanish so that
  the norm of $u$ is the maximal modulus of its coefficients? 
\end{Q}

In our answer to Question~\ref{q2}, $\Sch^p$ and Schur multiplication
are treated as a noncommutative analogue to $\Ell^p$ and convolution.
The main step is a careful study of the Schatten-von-Neumann norm
$\|x\|=\bigl(\tr(x^*x)^{p/2}\bigr)^{1/p}$ for $p$ an even integer.
The rule of matrix multiplication provides an expression for this norm
as a series in the matrix coefficients of $x$ and their complex
conjugate, indexed by the $p$uples $(v_1,v_2,\dots,v_p)$ satisfying
$(v_{2i-1},v_{2i}),(v_{2i+1},v_{2i})\in I$, where $v_{p+1}=v_1$: see the computation in Eq.~\eqref{compu}. These are best understood
as closed walks of length $p$ on the bipartite graph $G$ canonically
associated to $I$: its vertex classes are $\Col$ and $\Row$ and its
edges are given by the couples in $I$. A structure theorem for closed
walks and a detailed study of the particular case in which $G$ is a cycle yield
the two following theorems that answer Questions~\ref{q1} and~\ref{q2}.

\begin{thm}\label{th4}
  Let $p\in(0,\infty]\setminus\{2,4,6,\dots\}$. If the
  sequence of elementary matrices $(\e_{rc})_{(r,c)\in I}$ is a real
  1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$, then the graph
  $G$ associated to $I$ contains no
  cycle. In this case, $I$ is even a set of $\Varopoulos$-interpolation with
  constant 1: every sequence $\varphi\in\ell^\infty_I$ may be
  interpolated by a tensor
  $u\in\ell_\Col^\infty\projtens\ell_\Row^\infty$ such that
  $\|u\|=\|\varphi\|$.
\end{thm}

\begin{thm}\label{th5}
  Let $p\in\{2,4,6,\dots\}$. The sequence
  $(\e_{rc})_{(r,c)\in I}$ is a complex, or real, 1-un\-con\-di\-tio\-nal
  basic sequence in $\Sch^p$ if and only if $G$ contains no cycle of
  length $4,6,\dots,p$.
\end{thm}

These theorems hold also for the complete counterparts to
1-un\-con\-di\-tio\-nal basic sequences in the sense of
Def.~\ref{unc:def}$(c)$.

In particular, if we denote by $\mathrm U_p$ the property that
$(\e_{rc})_{(r,c)\in I}$ is a 1-un\-con\-di\-tio\-nal basic sequence in
$\Sch^p$, then we obtain the following hierarchy:
\[\mathrm U_p\text{ for a $p\in(0,\infty]\setminus\{2,4,6,\dots\}$}\
\Rightarrow\ \cdots\ \Rightarrow\ \mathrm U_{2n+2}\ \Rightarrow\ 
\mathrm U_{2n}\ \Rightarrow\ \cdots\ \Rightarrow\ \mathrm U_2\ .\]

If $\Col$ and $\Row$ are finite, extremal graphs without cycles of
given lengths remain an ongoing area of research in graph theory.
Finite geometries seem to provide all known examples of such graphs
when $\Col$ and $\Row$ become large. Proposition~\ref{extr:prp} and
Remark~\ref{extr:rem} gather up known facts on this issue.

One may also avoid the terminology of graph theory and give an answer
in terms of polygons drawn in a matrix by joining matrix coordinates
with sides that follow alternately the row (horizontal) and the column
(vertical) direction of the matrix:

\begin{itemize}
\item Suppose that $p$ is not an even integer. If a pattern $I$
  contains the vertices of such a polygon, then there is an operator
  $x\in \Sch^p$ whose matrix coefficients vanish outside $I$ and
  whose norm depends on the sign of its matrix coefficients. This
  condition is also necessary.
\item If matrix coordinates of nonzero matrix coefficients of $x$ are
  the vertices of such a polygon with $n$ sides, then the norm of $x$ in
  $\Sch^p$ depends on the argument of its matrix coefficients for
  every even integer $p\ge n$; if the matrix coefficients of $x$ are real, then
  the norm of $x$ even depends on the sign of its matrix coefficients.
  These conditions are also necessary. 
\end{itemize}

An elementary example is given by the set
\begin{equation}
  I=\bigl\{(r,c)\in\Z/7\Z\times\Z/7\Z:r+c\in\{0,1,3\}\bigr\}.
\label{steiner}
\end{equation}
The associated bipartite graph is known as the Heawood graph (Fig.~\ref{fig:1}:) it is
the incidence graph of the Fano plane (the finite projective plane
$\mathrm{PG}(2,2)$,) which is the smallest generalised triangle, and
corresponds to the Steiner system $\mathrm S(2,3;7)$. It contains no
cycle of length 4, but every pair of vertices is contained in a
cycle of length 6. 
\begin{figure}[ht]
  \includegraphics{heawood}\centering
  \caption{The Heawood graph}
  \label{fig:1}
\end{figure}

Thus the $p$-trace norm of every matrix with
pattern
\[
\bordermatrix{
  &\scriptstyle0&\scriptstyle1&\scriptstyle2&\scriptstyle3&\scriptstyle4&\scriptstyle5&\scriptstyle6\cr
  \scriptstyle0&*&*&0&*&0&0&0\cr
  \scriptstyle1&*&0&*&0&0&0&*\cr
  \scriptstyle2&0&*&0&0&0&*&*\cr
  \scriptstyle3&*&0&0&0&*&*&0\cr
  \scriptstyle4&0&0&0&*&*&0&*\cr
  \scriptstyle5&0&0&*&*&0&*&0\cr
  \scriptstyle6&0&*&*&0&*&0&0\cr}
\]
does not depend on the sign of its coefficients
if and only if $p\in\{2,4\}$. 

These results give a complete description of the situation in which
$(\e_{rc})_{(r,c)\in I}$ is a 1-un\-con\-di\-tio\-nal basis of the space
$\Sch^p_I$ it spans in $\Sch^p$. If this is not the case, $\Sch^p_I$ might
still admit some other 1-un\-con\-di\-tio\-nal basis. This leads to the
following more general question. 

\begin{Q}\label{q4}
  For which sets $I$ does $\Sch^p_I$ admit some kind of almost
  1-un\-con\-di\-tio\-nal finite dimensional 
expansion of the identity?
\end{Q}
                                
The metric unconditional approximation property $(muap)$ provides a
formal definition for the object of Question~\ref{q4}: see
Def.~\ref{muap:def}. We obtain the following results.

\begin{thm}
  Let $p\in[1,\infty]\setminus\{2,4,6,\dots\}$. If $\Sch^p_I$ has
  real $(muap)$, then the distance of any two vertices that are not in
  the same vertex class is asymptotically infinite in $G$: their
  distance becomes arbitrarily large by deleting a finite number of
  edges from $G$.
\end{thm}

\begin{thm}
  Let $p\in\{2,4,6,\dots\}$. The space $\Sch^p_I$ has complex, or
  real, $(muap)$ if and only if any two vertices at distance $2j+1\le
  p/2$ are asymptotically at distance at least $p-2j+1$.
\end{thm}

We now turn to a detailed description of this article. In
Section~\ref{ss2}, we 
provide tools for the computation of Schur
multiplier norms. Section~\ref{ss3} characterises idempotent Schur
multipliers and $0,1$-tensors in $\ell_\Col^\infty\projtens\ell_\Row^\infty$
of norm 1. In Section~\ref{ss4}, we define the complex and real
unconditional constants of basic sequences of elementary matrices and
show that they are not equal in general. Section~\ref{V} looks back on
Varopoulos' results about tensor algebras over discrete spaces.
Section~\ref{ss6} puts the connection between $p$-trace norm and
closed walks of length $p$ in the concrete form of closed walk
relations. In Section~\ref{ss7}, we compute the norm of relative Schur
multipliers by signs in the case that $G$ is a cycle, and estimate the
corresponding unconditional constants. Section~\ref{ss8} is dedicated
to a proof of Th.~\ref{th4} and an answer to Question~\ref{q3}.
Section~\ref{ss9} establishes Th.~\ref{th5}. In Section~\ref{ss10},
we study the metric unconditional approximation property for spaces
$\Sch^p_I$. The final section provides four kinds of examples: sets
obtained by a transfer of $n$-independent subsets of a discrete abelian
group, Hankel sets, Steiner systems and Tits' generalised polygons. 

\begin{Ter}
  $\Col$ is the set of \emph{columns} and $\Row$ is the set of
  \emph{rows}, both finite or countable and if necessary indexed by
  natural numbers. $V$, the set of \emph{vertices}, is their
  disjoint union $\Col\amalg\Row$: if there is a risk of confusion, an
  element $n\in V$ that is a column (vs.~a row) will be
  referred to as ``col~$n$'' (vs.~``row~$n$''.) An
  \emph{edge} on $V$ is a pair $\{v,w\}\subseteq V$. A
  \emph{graph} on $V$ is given by a set of edges $E$. A
  \emph{bipartite} graph on $V$ with vertex classes $\Col$ and
  $\Row$ has only edges $\{r,c\}$ such that $c\in\Col$ and $r\in\Row$
  and may therefore be given alternatively by the set of couples
  $I=\bigl\{(r,c)\in\Row\times\Col:\{r,c\}\in E\bigr\}$: this will be
  our custom throughout the article. A bipartite graph on $V$ is
  \emph{complete} if its set of couples $I$ is the whole of $R\times
  C$. Two graphs are \emph{disjoint} if so are the sets of vertices
  of their edges. $I$ is a \emph{column section} if
  $(r,c),(r',c)\in I\Rightarrow r=r'$, and a \emph{row section} if $(r,c),(r,c')\in
  I\Rightarrow c=c'$.
  
  A \emph{walk} of length $s\ge0$ in a graph is a sequence
  $(v_0,\dots,v_s)$ of $s+1$ vertices such that
  $\{v_0,v_1\}$, $\ldots\,$,~$\{v_{s-1},v_s\}$ are edges of the graph. A walk
  is a \emph{path} if its vertices are pairwise distinct. The
  \emph{distance} of two vertices in a graph is the minimal length
  of a path in the graph that joins the two vertices; it is infinite
  if no such path exists. A \emph{closed} walk of length $p\ge0$ in
  a graph is a sequence $(v_1,\dots,v_p)$ of $p$ vertices such that
  $\{v_1,v_2\},\dots,\{v_{p-1},v_p\},\{v_p,v_1\}$ are edges of the
  graph. Note that $p$ is necessarily even if the graph is bipartite.
  A closed walk is a \emph{cycle} if its vertices are pairwise
  distinct. We take the convention that if a closed walk in a
  bipartite graph on $V=\Col\amalg\Row$ is nonempty, then its first
  vertex is a column vertex: $v_1\in\Col$. We shall identify a path
  and a cycle with its set of edges $\{r,c\}$ or the corresponding
  set of couples $(r,c)$.
  
  A bipartite graph on $V$ is a \emph{tree} if there is exactly one
  path between any two of its vertices. In this case, its vertices may
  be indexed by finite words over its set of vertices in the following
  way. Choose any row vertex $r$ as \emph{root} and index it by $\emptyset$. If $v$ is a vertex
  and $(r,c,\dots,v)$ is the unique path from
  $r$ to $v$, let the word $c\concat\cdots\concat v$ index $v$. Let
  $W$ be the set of all words thus formed. Then
  \begin{itemize}
  \item $\emptyset\in W$ and every beginning of a word in $W$ is also in
    $W$: if $w\in W\setminus\{\emptyset\}$, then $w$ is the
    concatenation $w'\concat v$ of a word $w'\in W$ with a letter $v$;
  \item words of even length index row vertices;
  \item words of odd length index column vertices;
  \item a pair of vertices is
    an edge exactly if their indices have the form $w$ and $w\concat
    v$, where $w$ is a word and $v$ is a letter.
  \end{itemize}
  A \emph{forest} is a union of pairwise disjoint trees; equivalently,
  it is a cycle free graph.
\end{Ter}

\begin{Not}
  Let $\T=\{z\in\C:|z|=1\}$. 
  
  The unit ball of a Banach space $X$ is denoted by $B_X$.

  Given an index set
  $I$ and $q\in I$, $\e_q$ is the sequence defined on $I$ as the
  indicator function $\chi_{\{q\}}$ of the singleton $\{q\}$.

  Let $I=\Row\times\Col$ and $q=(r,c)$. Then $\e_q=\e_{rc}$ 
  is the \emph{elementary matrix} identified with the operator from
  $\ell^2_\Col$ to $\ell^2_\Row$ that maps $\e_c$ on $\e_r$ and all
  other basis vectors on 0. The \emph{matrix coefficient} at
  coordinate $q$ of an operator $x$ from $\ell^2_\Col$ to $\ell^2_\Row$
  is $x_q=\tr\e_q^*x$ and its \emph{matrix representation} is
  $(x_q)_{q\in\Row\times\Col}=\sum_{q\in\Row\times
    \Col}x_q\e_q$. The \emph{support} of $x$ is $\{q\in\Row\times\Col:x_q\ne0\}$.
  
  The \emph{Schatten-von-Neumann class} $\Sch^p$, $0<p<\infty$,
  is the space of those compact operators $x$ from $\ell^2_\Col$ to
  $\ell^2_\Row$ such that
  $\|x\|_p^p=\tr|x|^p=\tr(x^*x)^{p/2}<\infty$.
  $\Sch^\infty$ is the space of compact operators with the
  operator norm. $\Sch^p$ is a quasi-normed space, and a Banach
  space if $p\ge1$. Let $(\Row_n\times\Col_n)_{n\ge0}$ be a sequence
  of finite sets that tends to $\Row\times\Col$. Then the sequence
  of operators $P_n\colon x\mapsto\sum_{q\in\Row_n\times
    \Col_n}x_q\e_q$ tends pointwise to the identity on
  $\Sch^p$ if $p\ge1$. 
  
  For $I\subseteq\Row\times\Col$, the \emph{entry space}
  $\Sch^p_I$ is the subspace of those $x\in \Sch^p$ whose
  support is a subset of $I$. $\Sch^p_I$ is also the closed
  subspace of $\Sch^p$ spanned by $(\e_q)_{q\in I}$.
  
  The $\Sch^p$-valued Schatten-von-Neumann class
  $\Sch^p(\Sch^p)$ is the space of those compact operators
  $x$ from $\ell^2_\Col$ to $\ell^2_\Row(\Sch^p)$ such that
  $\|x\|_p^p=\tr(\tr|x|^p)<\infty$, where the inner trace is the
  $\Sch^p$-valued analogue of the usual trace: such operators
  have an $\Sch^p$-valued matrix representation and their support
  is defined as in the scalar case. An element
  $x\in\Sch^p(\Sch^p)$ can also be considered as a compact
  operator from $\ell^2_\Col(\ell_2)=\ell_2\otimes_2\ell^2_\Col$ to
  $\ell^2_\Row(\ell_2)=\ell_2\otimes_2\ell^2_\Row$ such that
  $\|x\|_p^p=\tr\otimes\tr|x|^p<\infty$; the matrix coefficient
  of $x$ at $q$ is then $x_q = (\Id_{\Sch^p}\otimes\tr)
  \bigl((\Id_{\ell_2}\otimes\e_q^*)x\bigr)$ and its matrix
  representation is $\sum_{q\in\Row\times\Col}x_q\otimes\e_q$. The
  entry space $\Sch^p_I(\Sch^p)$ is defined in the same way
  as $\Sch^p_I$.
  
  A \emph{relative Schur multiplier} on $\Sch^p_I$ is a
  sequence $\varphi=(\varphi_q)_{q\in I}\in\C^I$ such that the
  associated \emph{Schur multiplication operator} $\Schur_\varphi$
  defined by $\e_q\mapsto\varphi_q\e_q$ for $q\in I$ is bounded on
  $\Sch^p_I$. The Schur multiplier $\varphi$ is furthermore
  \emph{completely bounded} (c.b.~for short) on $\Sch^p_I$ if
  $\Id_{\Sch^p}\otimes\Schur_\varphi$, the operator defined by
  $x_q\e_q\mapsto\varphi_qx_q\e_q$ for $x_q\in \Sch^p$ and $q\in
  I$, is bounded on $\Sch^p_I(\Sch^p)$ (see
  \cite[Lemma~1.7]{pi98}.) The norm of $\varphi$ is the norm of
  $\Schur_\varphi$ and its \emph{complete norm} is the norm of
  $\Id_{\Sch^p}\otimes\Schur_\varphi$.  This norm is the supremum
  of the norm of its restrictions to finite rectangle sets
  $\Row'\times\Col'$. Note that $\varphi$ is a Schur multiplier
  on $\Sch^\infty$ if and only if, for every bounded operator
  $x\colon\ell^2_\Col\to\ell^2_\Row$, $(\varphi_qx_q)$ is the matrix representation
  of a bounded operator; also 
  $\varphi$ is automatically c.b.~on $\Sch^\infty$
  \cite[Th.~5.1]{pi01}. We used \cite{pi98,pi01} as a reference.
  
  Let $G$ be a compact abelian group endowed with its normalised Haar
  measure. Let $\Gamma=\hat G$ be the dual group of characters on $G$.
  The \emph{spectrum} of an integrable function $f$ on $G$ is
  $\{\gamma\in\Gamma:\hat f(\gamma)\ne0\}$. Let
  $\Lambda\subseteq\Gamma$. If $X$ is a space of integrable functions
  on $G$, then $X_\Lambda$ is the \emph{translation invariant}
  subspace of those $f\in X$ whose spectrum is a subset of $\Lambda$.
  
  Let $X$ be the space of continuous functions $\Cont(G)$ or the
  Lebesgue space $\Ell^p(G)$ with $0<p<\infty$. Then $X_\Lambda$ is
  also the closed subspace of $X$ spanned by $\Lambda$.
  A \emph{relative Fourier multiplier} on $X_\Lambda$ is a sequence
  $\mu=(\mu_\gamma)_{\gamma\in\Lambda}\in\C^\Lambda$ such that the
  associated convolution operator $\Fourier_\mu$ defined by
  $\gamma\mapsto\mu_\gamma\gamma$ for $\gamma\in\Lambda$ is bounded on
  $X_\Lambda$. The Fourier multiplier $\mu$ is furthermore c.b.\ if
  $\Id_{\Sch^p}\otimes\Fourier_\mu$, the operator defined by
  $a_\gamma\gamma\mapsto \mu_\gamma a_\gamma\gamma$ for
  $a_\gamma\in\Sch^p$ and $\gamma\in\Lambda$, is bounded on the
  $\Sch^p$-valued space $X_\Lambda(\Sch^p)$ (where
  $p=\infty$ if $X=\Cont(G)$.) The norm of
  $\mu$ is the norm of $\Fourier_\mu$ and its complete norm is the
  norm of $\Id_{\Sch^p}\otimes\Fourier_\mu$. Note that $\mu$ is a
  Fourier multiplier on $\Cont_\Lambda(G)$ if and only if, for
  every $f\in\Ell_\Lambda^\infty(G)$, $\sum\mu_\gamma\hat
  f(\gamma)\gamma$ is the Fourier series of an element of
  $\Ell_\Lambda^\infty(G)$: $\mu$ is a
  \emph{relative Fourier multiplier on $\Ell^\infty(G)$}; also $\mu$ is automatically c.b.~on
  $\Cont_\Lambda(G)$ \cite[Cor.\ 3.18]{pi01}.

  Let $X,Y$ be Banach spaces and $u\in X\otimes Y$. Its \emph{projective}
  tensor norm is
  \begin{equation*}
    \|u\|_{X\projtens Y}=\inf\biggl\{\sum_{j=1}^n\|x_j\|\,\|y_j\|:u=\sum_{j=1}^nx_j\otimes
    y_j\biggr\}
  \end{equation*}
  and $X\projtens Y$ is the completion of $X\otimes Y$ with respect to
  this norm. Note that $\ell_\infty^n\projtens \ell_\infty^m\subset
  \cnaught\projtens\cnaught$ because $\ell_\infty^n$ and
  $\ell_\infty^m$ are 1-complemented in $\cnaught$, and that
  $\cnaught\projtens\cnaught\subset\ell_\infty\projtens\ell_\infty$
  because $\ell_\infty$ is the bidual of $\cnaught$. 
  
  Let $\sum x_j\otimes y_j$ be any representation of the tensor $u$.
  If $\xi\otimes\eta\in X^*\otimes Y^*$, we define
  $\langle\xi\otimes\eta,u\rangle=\sum\langle\xi,x_j\rangle\langle\eta,y_j\rangle$.
  The \emph{injective} tensor norm of $u$ is
  \begin{equation*}
    \|u\|_{X\injtens Y} =
    \sup_{(\xi,\eta)\in B_{X^*}\times B_{Y^*}}|\langle\xi\otimes\eta,u\rangle|
  \end{equation*}
  and $X\injtens Y$ is the completion of $X\otimes Y$ with
  respect to this norm. 

  If $X$ and $Y$ are both finite dimensional, then
  \begin{equation*}
    \bigl(X\injtens Y\bigr)^*=X^*\projtens Y^*\quad\text{and}\quad
    \bigl(X\projtens Y\bigr)^*=X^*\injtens Y^*.
  \end{equation*}
  Further
  $\bigl(\cnaught\projtens\cnaught\bigr)^*=\ell_1\injtens\ell_1$: in
  fact, $\bigl(\cnaught\projtens\cnaught\bigr)^*$ may be identified
  with the space of bounded operators from $\cnaught$ to $\ell_1$ and
  $\ell_1\injtens\ell_1$ may be identified with the closure of finite
  rank operators in that space, and they are the same because every
  bounded operator from $\cnaught$ to $\ell_1$ is compact and $\ell_1$
  has the approximation property. 

  If $X$ is a sequence space on
  $\Col$ and $Y$ is a sequence space on $\Row$, then the
  \emph{coefficient} of the tensor $u$ at $(r,c)$ is
  $\langle\e_c\otimes\e_r,u\rangle$. Its \emph{support} is the set
  of coordinates $(r,c)$ of its nonvanishing coefficients. One may use
  \cite{ry02} as a reference.
  
\end{Not}

\section{Relative Schur multipliers}\label{ss2}

\noindent The following proposition is a straightforward consequence of
\cite{pps89}.

\begin{prp}\label{pinf}
  Let $I\subseteq\Row\times\Col$ and $\varphi$ be a Schur multiplier
  on $\Sch^\infty_I$ with norm $D$. Then $\varphi$ is also a c.b.~Schur multiplier
  on $\Sch^p_I$ for every $p\in(0,\infty]$, with complete norm bounded by $D$.
\end{prp}

\begin{proof}
  We may assume that $D=1$. Let $\Row'\times\Col'$ be any finite subset of
  $\Row\times\Col$. By \cite[Th.~3.2]{pps89}, there exist vectors
  $w_c$ and $v_r$ of norm at most 1 in a Hilbert space
  $H$ such that $\varphi_{rc}=\langle w_c,v_r\rangle$ for every
  $(r,c)\in I\cap\Row'\times\Col'$. If we define $W\colon\ell^2_{\Col'}\to\ell^2_{\Col'}(H)$ and
  $V\colon\ell^2_{\Row'}\to\ell^2_{\Row'}(H)$ by
  $W\zeta=(\zeta_cw_c)_{c\in\Col'}$ and $V\eta=(\eta_rv_r)_{r\in\Row'}$,
  then $V$ and $W$ have norm at most 1, and the proposition follows from the factorisation
  \begin{equation*}
    \Schur_\varphi x=V^*(x\otimes\Id_H)W
  \end{equation*}
  for every $x$ with support in $I\cap\Row'\times\Col'$.
\end{proof}

\begin{rem}
  \'Eric Ricard showed us an elementary proof
  that a Schur multiplier on $\Sch^\infty_I$ is automatically
  c.b., included here by his kind permission. A Schur multiplier
  $\varphi$ is bounded on $\Sch^\infty_I$ by a constant $D$ if and only if
  \begin{equation}\label{b}
    \forall\,\xi\in
    B_{\Sch^\infty_I}~\forall\,\eta\in B_{\ell^2_\Row}~\forall\,\zeta\in
    B_{\ell^2_\Col}\quad\biggl|\sum_{(r,c)\in
      I}\eta_r\varphi_{rc}\xi_{rc}\zeta_c\biggr|\le D.
  \end{equation}
  It is furthermore completely bounded on $\Sch^\infty_I$ by $D$ if
  \begin{equation}\label{cb}
    \forall\,x\in
    B_{\Sch^\infty_I(\Sch^\infty)}~\forall\,y\in B_{\ell^2_\Row(\ell_2)}~\forall\,z\in
    B_{\ell^2_\Col(\ell_2)}\quad\biggl|\sum_{(r,c)\in I}\varphi_{rc}\langle
    y_r,x_{rc}z_c\rangle\biggr|\le D.
  \end{equation}
  Suppose that $x,y,z$ are as quantified in Ineq.~\eqref{cb}. Let
  \[
  \xi_{rc}=\langle y_r/\|y_r\|,x_{rc}z_c/\|z_c\|\rangle ,~
  \eta_r=\|y_r\|_{\ell_2} ~\text{and}~ \zeta_c=\|z_c\|_{\ell_2}.
  \] 
  Then
  $\|\eta\|_{\ell^2_\Row},\|\zeta\|_{\ell^2_\Col}\le1$ and
  \begin{multline*}
    \|\xi\|=\sup\biggl\{\biggl| \sum_{(r,c)\in I} \langle \alpha_ry_r/\|y_r\|_{\ell_2},
    x_{rc}\beta_c z_c/\|z_c\|_{\ell_2} \rangle \biggr| : \alpha\in B_{\ell^2_\Row},\,
    \beta\in B_{\ell^2_\Col} \biggr\}\\
    \le \|x\| 
    \sup_{\alpha\in B_{\ell^2_\Row}}
    \bigl\|\bigl(\alpha_ry_r/\|y_r\|_{\ell_2}\bigr)\bigr\|_{\ell^2_\Row(\ell_2)}
    \sup_{\beta\in B_{\ell^2_\Col}}
    \bigl\|\bigl(\beta_cz_c/\|z_c\|_{\ell_2}\bigr)\bigr\|_{\ell^2_\Col(\ell_2)}\le 1, 
  \end{multline*}
  so that Ineq.~\eqref{b} implies Ineq.~\eqref{cb}.
\end{rem}

The fact that the canonical basis of an $\ell^2$ space is
1-un\-con\-di\-tio\-nal yields that Schatten-von-Neumann norms are
\emph{matrix unconditional} in the terminology of \cite{sc78}:
\begin{equation}
  \label{mu}
  \forall\,\zeta\in\T^\Col~\forall\,\eta\in\T^\Row\quad
  \Bigl\|\sum_{(r,c)\in\Row\times\Col}\zeta_c\eta_ra_{rc}\e_{rc}\Bigr\|_p=
  \Bigl\|\sum_{(r,c)\in\Row\times\Col}a_{rc}\e_{rc}\Bigr\|_p
\end{equation}
for every finitely supported sequence of complex or
$\Sch^p$-valued coefficients $a_{rc}$. Let $\zeta\otimes\eta$
denote the \emph{elementary} Schur multiplier
$(\zeta_c\eta_r)_{(r,c)\in\Row\times\Col}$. Equation~\eqref{mu} shows that if
$\zeta\in\T^{\Col}$ and $\eta\in\T^{\Row}$, then
$\Schur_{\zeta\otimes\eta}$ is an isometry on every $S^p$. This yields
that if 
$\zeta\in\ell^\infty_\Col$, $\eta\in\ell^\infty_\Row$, then the
complete norm of $\Schur_{\zeta\otimes\eta}$ is 
$\|\zeta\|_{\ell^\infty_\Col}\|\eta\|_{\ell^\infty_\Row}$ on every
$\Sch^p$. 

Relative Schur multipliers also have a central place among operators
on $\Sch^p_I$ because they appear as the range of a contractive
projection defined by the following averaging scheme.

\begin{dfn}
  Let $T\colon\Sch^p_J\to\Sch^p_I$ be an operator. Let
  $\Row'\times\Col'$ be a finite subset of $\Row\times\Col$ and let
  $P_{\Row'\times\Col'}$ be the contractive projection onto $\Sch^p_{\Row'\times\Col'}$
  defined by the Schur multiplier
  $\chi_{\Col'}\otimes\chi_{\Row'}$. Then the \emph{average of $T$
    with respect to $\Row'\times\Col'$} is given by
  \begin{equation}\label{ave}
    [T]_{\Row'\times\Col'}(x) = \int_{\T^{\Row}}\mathrm{d}\eta\,
    \int_{\T^{\Col}}\mathrm{d}\zeta\,
    \Schur_{\zeta^*\otimes\eta^*}
    P_{\Row'\times\Col'}T(\Schur_{\zeta\otimes\eta}x),
  \end{equation}
  where $\zeta^*=(\overline{\zeta_c})_{c\in\Col}$ and
  $\eta^*=(\overline{\eta_r})_{r\in\Row}$. 
\end{dfn}

\begin{prp}\label{aver}
  Let $T\colon\Sch^p_J\to\Sch^p_I$ be an operator and
  $\Row'\times\Col'$ a finite subset of $\Row\times\Col$. Then
  $[T]_{\Row'\times\Col'}$ is a Schur multiplication operator from
  $\Sch^p_J$ to $\Sch^p_{I\cap\Row'\times\Col'}$ such that
  $\|[T]_{\Row'\times\Col'}\|\le\|T\|$. In fact,
  $[T]_{\Row'\times\Col'}=M_{\varphi^{\Row'\times\Col'}}$ with 
  \begin{equation*}
    \varphi^{\Row'\times\Col'}_{rc}=
    \begin{cases}
      \tr\e_{rc}^*T(\e_{rc})&\text{if
        $(r,c)\in J\cap\Row'\times\Col'$}\\
      0&\text{if $(r,c)\in J\setminus\Row'\times\Col'$.}
    \end{cases}
  \end{equation*}
  If $T$ is a projection onto $\Sch^p_I$, then
  $\varphi^{\Row'\times\Col'}=\chi_{I\cap\Row'\times\Col'}$, so that
  $[T]_{\Row'\times\Col'}$ is a projection onto
  $\Sch^p_{I\cap\Row'\times\Col'}$. Let
  $\varphi=\bigl(\tr\e_q^*T(\e_q)\bigr)_{q\in J}$. Then
  $\|\Schur_\varphi\|\le\|T\|$ and we define the \emph{average of $T$} by $[T]=\Schur_\varphi$.  
\end{prp}

\begin{proof}
  Formula~\eqref{ave} shows that
  $\|[T]_{\Row'\times\Col'}(x)\|\le\|T\|\,\|x\|$. We have
  \begin{align*}
    [T]_{\Row'\times\Col'}(\e_{rc}) &=
    \int_{\T^{\Row}}\mathrm{d}\eta\, \int_{\T^{\Col}}\mathrm{d}\zeta\,
    \Schur_{\zeta^*\otimes\eta^*}
    P_{\Row'\times\Col'}T(\zeta_c\eta_r\e_{rc})\\
    &= \int_{\T^{\Row}}\mathrm{d}\eta\,
    \int_{\T^{\Col}}\mathrm{d}\zeta\, \zeta_c\eta_r
    \Schur_{\zeta^*\otimes\eta^*} \sum_{(r',c')\in\Row'\times\Col'}
    \tr\bigl(\e_{r'c'}^*T(\e_{rc})\bigr) \e_{r'c'}\\
    &= \sum_{(r',c')\in\Row'\times\Col'}
    \int_{\T^{\Row}}\mathrm{d}\eta\, \int_{\T^{\Col}}\mathrm{d}\zeta\,
    \zeta_c\eta_r \tr\bigl(\e_{r'c'}^*T(\e_{rc})\bigr)
    \zeta_{c'}^{-1}\eta_{r'}^{-1}\e_{r'c'}
    = \varphi^{\Row'\times\Col'}_{rc}\e_{rc}.
  \end{align*}
  As the norm of a Schur multiplier is the supremum of the norm of its
  restrictions to finite rectangle sets, this shows that
  $\varphi$ is a Schur multiplier on $\Sch^p_J$ and
  $\|\Schur_\varphi\|\le\|T\|$. 
  If $T$ is a projection onto $\Sch^p_I$, note that
  $\tr\e_{rc}^*T(\e_{rc})=\chi_I(r,c)$.
\end{proof}

The following proposition relates Fourier multipliers to Herz-Schur
multipliers in the fashion of \cite[Th.~6.4]{pi01} and will be very useful
in the exact computation of the norm of certain relative Schur multipliers.

\begin{prp}\label{transfer}
  Let $\Gamma$ be a countable discrete abelian group and
  $\Lambda\subseteq\Gamma$. Let $\Row$ and $\Col$ be two copies of
  $\Gamma$ and consider $I=\{(r,c)\in\Row\times\Col:r-c\in\Lambda\}$.
  Let $\varphi\in\C^I$ such that there is $\mu\in\C^\Lambda$ with
  $\varphi(r,c)=\mu(r-c)$ for all $(r,c)\in I$. Let $G=\hat\Gamma$, so
  that $\Gamma$ is the group of characters on the compact group
  $G$. Let $p\in(0,\infty]$. 
  \begin{enumerate}
  \item The complete norm of the relative Schur multiplier $\varphi$
    on $\Sch^p_I$ is bounded by the complete norm of the relative
    Fourier multiplier $\mu$ on
    $\Ell^p_\Lambda(G)$. 
  \item Suppose that $\Gamma$ is finite. The norm of the relative Fourier multiplier $\mu$ on
    $\Ell^p_\Lambda(G)$ is bounded by the norm of the relative Schur
    multiplier $\varphi$ on $\Sch^p_I$. The same holds for
    complete norms.
  \end{enumerate}
\end{prp}

\begin{rem}
  Part~$(b)$ is just an abstract counterpart to \cite[Chapter~6,
  Lem\-ma~3.8]{pe03}, where the case of the finite cyclic group
  $\Gamma=\Z/n\Z$ is treated.
\end{rem}

\begin{proof}$(a)$ is \cite[Lemma 8.1.4]{pi98}:
   for all $a_q\in \Sch^p$, of which only a finite number
   are nonzero, and all $g\in G$, we have by matrix unconditionality
   (Eq.~\eqref{mu})
   \begin{multline}\label{tr1}
     \Bigl\|\sum_{q\in I}a_q\e_q\Bigr\|_{\Sch^p_I(\Sch^p)} = \Bigl\|\sum_{(r,c)\in
       I}r(g)c(g)^{-1}a_{rc}\e_{rc}\Bigr\|_{\Sch^p_I(\Sch^p)}\\
     = \biggl\| \sum_{\gamma\in\Lambda}
     \biggl(\sum_{\substack{(r,c)\in I\\r-c=\gamma}}
     a_{rc}\e_{rc}\biggr)\gamma(g)\biggr\|_{\Sch^p_I(\Sch^p)} =
     \biggl\|\sum_{\gamma\in\Lambda} \biggl(\sum_{\substack{(r,c)\in
         I\\r-c=\gamma}}
     a_{rc}\e_{rc}\biggr)\gamma\biggr\|_{\Ell^p_\Lambda(G,\Sch^p(\Sch^p))}.
   \end{multline}
   This yields an isometric embedding of $\Sch^p_I(\Sch^p)$
   in $\Ell^p_\Lambda(G,\Sch^p_I(\Sch^p))$. As
   $\Sch^p(\Sch^p)$ may be identified with 
   $\Sch^p(\ell^2_\Gamma(\ell^2))$,
   \begin{equation*}\label{tr2}
     \Bigl\| \sum_{q\in I} \varphi_qa_q\e_q
     \Bigr\|_{\Sch^p_I(\Sch^p)} = \biggl\|
     \sum_{\gamma\in\Lambda} \mu_\gamma \biggl(
     \sum_{\substack{(r,c)\in I\\r-c=\gamma}} a_{rc}\e_{rc} \biggr)
     \gamma \biggr\|_{\Ell^p_\Lambda(G,\Sch^p(\Sch^p))}
     \le \| \Id\otimes\Fourier_\mu \| \Bigl\| \sum_{q\in I} a_q\e_q
     \Bigr\|_{\Sch^p_I(\Sch^p)}.
   \end{equation*}

$(b)$.  Let us embed $\Ell^p_\Lambda(G)$ into $\Sch^p_I$ by
$f\mapsto\fourier_{\hat f}$, where
$\fourier_{\hat f}\colon\ell^2_\Col\to\ell^2_\Row$ is the convolution operator defined by 
\[\fourier_{\hat f}\e_c
=\hat f*\e_c
=\sum_{\gamma\in\Lambda}\hat f(\gamma)\e_\gamma*\e_c
=\sum_{r-c\in\Lambda}\hat f(r-c)\e_r:\]
$\fourier_{\hat f}$ has the matrix representation $\sum_{(r,c)\in I}\hat
f(r-c)\e_{rc}$. The characters $g\in G$ form an 
orthonormal basis for $\ell^2_\Col$ such that $\fourier_{\hat f}g=f(g)g$: 
therefore 
\[{\|\fourier_{\hat f}\|}_p=\Bigr(\sum_{g\in
  G}|f(g)|^p\Bigr)^{1/p}=(\card G)^{1/p}{\|f\|}_{\Ell^p(G)}.\] As
$\Schur_{\varphi}\fourier_{\hat f}=\fourier_{\widehat{\Fourier_\mu f}}$, this shows that
the norm of $\mu$ on $\Ell^p_\Lambda(G)$ is the norm of $\varphi$ on
the subspace of circulant matrices in $\Sch^p_I$. The same
holds for complete norms.
\end{proof}

\section{Idempotent Schur multipliers of norm 1}\label{ss3}

\noindent A Schur multiplier is \emph{idempotent} if it is the indicator
function $\chi_I$ of some set $I\subseteq\Row\times\Col$; if $\chi_I$
is a Schur multiplier on $\Sch^p$, then it is a projection of $\Sch^p$ onto
$\Sch^p_I$.  Idempotent Schur multipliers on $\Sch^p$ and tensors in
$\ell^\infty_\Col\projtens \ell^\infty_\Row$ with $0,1$ coefficients
of norm 1 may be characterised by the combinatorics of $I$.

\begin{prp}\label{complem}
  Let $I\subseteq\Row\times\Col$ be nonempty and $0<p\ne2<\infty$. The following
  are equivalent. 
  
  \begin{enumerate}
  \item For every finite rectangle set $\Row'\times\Col'$
    intersecting $I$
    \begin{equation*}
      \biggl\| \sum_{(r,c)\in I\cap\Row'\times\Col'} \e_c\otimes\e_r
      \biggr\| _{\ell^\infty_\Col\projtens \ell^\infty_\Row}=1.
    \end{equation*}
  \item $\Sch^p_I$ is completely 1-complemented in $\Sch^p$.
  \item $\Sch^p_I$ is 1-complemented in $\Sch^p$.
  \item $I$ is a union of pairwise disjoint complete bipartite graphs: there
    are pairwise disjoint sets $\Row_j\subseteq\Row$ and pairwise disjoint sets
    $\Col_j\subseteq\Col$ such that $I=\bigcup\Row_j\times\Col_j$.
  \end{enumerate}
  
\end{prp}

Property $(d)$ means that the pattern $I$ is, up to a permutation of columns and rows, block-diagonal:
\[
\bordermatrix{
  &\scriptstyle C_1&\scriptstyle C_2&\scriptstyle C_3&\scriptstyle\cdots\cr
  \scriptstyle R_1&*&0&0&\cdots\cr
  \scriptstyle R_2&0&*&0&\ddots\cr
  \scriptstyle R_3&0&0&*&\ddots\cr
  \scriptstyle\hfil\vdots&\vdots&\ddots&\ddots&\ddots\cr
  }\text.
\]
 

\begin{proof}
  $(b)\Rightarrow(c)$ is trivial.

  $(a)\Rightarrow(b)$.  The complete norm of a Schur multiplier $\varphi$ on
  $\Sch^p$ is the supremum of the complete norm of its restrictions
  $\varphi'=(\varphi_q)_{q\in\Row'\times\Col'}$ to finite rectangle
  sets $\Row'\times\Col'$. Furthermore, the complete norm of an elementary
  Schur multiplier $(\eta_c\zeta_r)_{(r,c)\in\Row\times\Col}=\eta\otimes\zeta$ on
  $\Sch^p$ equals
  $\|\eta\|_{\ell^\infty_\Col}\|\zeta\|_{\ell^\infty_\Row}$.
  
  $(c)\Rightarrow(d)$. If $I$ is not a union of pairwise disjoint
  complete bipartite graphs, then there are $r_0,r_1\in\Row$ and 
  $c_0,c_1\in\Col$ such that 
  \begin{equation*}
    I'=I\cap\{r_0,r_1\}\times\{c_0,c_1\}=\{(r_0,c_0),(r_1,c_0),(r_0,c_1)\}.
  \end{equation*}
  By Proposition~\ref{aver}, the average of a contractive projection of $\Sch^p$ onto
  $\Sch^p_I$ with respect to $\{r_0,r_1\}\times\{c_0,c_1\}$ would
  be the contractive projection associated to the Schur multiplier
  $\chi_{I'}$.  Let $x(t)$, $t\in\R$, be the
  operator from $\ell^2_C$ to $\ell^2_R$ whose matrix coefficients
  vanish except for its $\{r_0,r_1\}\times\{c_0,c_1\}$ submatrix,
  which is
  $\begin{pmatrix}1&\sqrt2\\
    \sqrt2&t\end{pmatrix}$. Its eigenvalues are
  \begin{equation*}
    \frac{1+t+\sqrt{9-2t+t^2}}2=2+\frac t3+o(t),~
    \frac{1+t-\sqrt{9-2t+t^2}}2=-1+\frac{2t}3+o(t),
  \end{equation*}
  so that
  \begin{equation*}
    \left\{
      \begin{aligned}
        &\|x(t)\|_\infty=2+t/3+o(t)\\
        &\|x(t)\|_p^p=2^p+1+p(2^p-4)t/6+o(t)\quad\text{for}~0<p<\infty
      \end{aligned}
    \right.
  \end{equation*}
  and therefore $\|\chi_{I'}*x(t)\|_p=\|x(0)\|_p>\|x(t)\|_p$ for some $t\ne0$ if
  $p\ne2$.
  
  $(d)\Rightarrow(a)$. Suppose $(d)$ and let $\Row'\times\Col'$
  intersect $I$. Then there are pairwise disjoint sets $\Row'_j$ and
  pairwise disjoint sets $\Col'_j$ such that $I\cap\Row'\times
  \Col'=\Row'_1\times\Col'_1\cup\dots\cup\Row'_n\times\Col'_n$ and
  \begin{equation*}
    \sum_{(r,c)\in I\cap\Row'\times\Col'}\e_c\otimes\e_r = 
    \sum_{j=1}^{n}\chi_{\Col'_j}\otimes\chi_{\Row'_j} = 
    \average_{\epsilon_j=\pm1}
    \biggl(\sum_{j=1}^{n}\epsilon_j\chi_{\Col'_j}\biggr)
    \otimes\biggl(\sum_{j=1}^{n}\epsilon_j\chi_{\Row'_j}\biggr)
  \end{equation*}
  which is an average of elementary tensors of norm 1, so that its
  projective tensor norm is bounded by 1, and actually is equal to
  1. 
\end{proof}

\begin{rem}
  Note that the proof of Prop.~\ref{complem} shows that the norm of a projection
  $\Schur_{\chi_I}\colon \Sch^\infty\to \Sch^\infty_I$ is either 1 or at least $2/\sqrt3$,
  as
  \begin{equation*}
    \left\|\begin{pmatrix}1&\sqrt2\\ \sqrt2&-1\end{pmatrix}
    \right\|_\infty=\sqrt3,\qquad
    \left\|\begin{pmatrix}1&\sqrt2\\ \sqrt2&0\end{pmatrix}\right\|_\infty=2.
  \end{equation*}
  This is a noncommutative analogue to the fact that an idempotent measure on a
  locally compact abelian group $G$ has either norm 1 or at least
  $\sqrt5/2$ \cite[Th.~3.7.2]{ru62}. The norm of $\Schur_{\chi_I}$ actually equals
  $2/\sqrt3$ for $I=\{(0,0),(0,1),(1,0)\}$, as shown in \cite[Lemma
  3]{liv95}. In fact, the following decomposition holds:
  \begin{multline*}
    \e_0\otimes\e_0+\e_0\otimes\e_1+\e_1\otimes\e_0=\\
    \bigl(
    (\e^{-\iu\pi/12},\e^{\iu\pi/4})\otimes
    (\e^{-\iu\pi/12},\e^{\iu\pi/4})+
    (\e^{\iu\pi/12},\e^{-\iu\pi/4})\otimes
    (\e^{\iu\pi/12},\e^{-\iu\pi/4})\bigr)/\sqrt3.
  \end{multline*}
\end{rem}

\begin{rem}
  Results related to the equivalence of $(c)$ with $(d)$ have been obtained
  independently by Banks and Harcharras \cite{bh04}.
\end{rem}

\section{\texorpdfstring{Unconditional basic sequences in $\Sch^p$}{Unconditional basic sequences of elementary matrices in Schatten classes}}

\label{ss4}

\begin{dfn}
  \label{unc:def}
  Let $0<p\le\infty$ and $I\subseteq\Row\times\Col$. Let $\Sign=\T$
  (vs.~$\Sign=\{-1,1\}$.) 
  
  \begin{enumerate}
  \item $I$ is an \emph{unconditional basic sequence} in $\Sch^p$ if there is a
    constant $D$ such that
    \begin{equation}\label{u}
      \Bigl\|\sum_{q\in I}\epsilon_qa_q\e_q\Bigr\|_p \le D
      \Bigl\|\sum_{q\in I} a_q\e_q\Bigr\|_p
    \end{equation}
    for every  choice of signs $\epsilon_q\in\Sign$
    and every finitely supported sequence of complex coefficients $a_q$. Its
    \emph{complex \klammer{vs.~real} unconditional constant} is the
    least such constant $D$. 
  \item $I$ is a \emph{completely} unconditional basic sequence in $\Sch^p$
    if there is a constant $D$ such that \eqref{u} holds for every choice of signs
    $\epsilon_q\in\Sign$ and every finitely supported sequence of operator
    coefficients $a_q\in \Sch^p$. Its complex (vs.~real) \emph{complete}
    unconditional constant is the least such constant $D$.
  \item $I$ is a complex
    (vs.~real, complex completely, real completely)
    \emph{1-un\-con\-di\-tio\-nal} basic sequence in $\Sch^p$ if its complex
    (vs.~real, complex complete, real complete) unconditional
    constant is 1: Inequality \eqref{u} turns into the equality
    \begin{equation*}
      \Bigl\|\sum_{q\in I}\epsilon_qa_q\e_q\Bigr\|_p =
      \Bigl\|\sum_{q\in I} a_q\e_q\Bigr\|_p.
    \end{equation*}
  \end{enumerate}
  
\end{dfn}

If Inequality~\eqref{u} holds for every real choice of signs, then it
also holds for every complex choice of signs at the cost of replacing
$D$ by $D\pi/2$ (see \cite{se97},) so that there is no need to
distinguish between complex and real unconditional basic sequences.

The notions defined in $(a)$ and $(b)$ are called \emph{$\sigma(p)$ sets} and
\emph{complete $\sigma(p)$ sets} in \cite[\S4]{ha98} and \cite{hno01} (see
also the survey \cite[\S\,9]{px03}.) The notions defined in $(c)$ are
their isometric counterparts.

By \cite[proof of Cor.~4]{sc78}, the real unconditional constant of
any basis of $\Sch^p_I$ cannot be lower than a fourth of the real
unconditional constant of $I$ in $\Sch^p$.

\begin{exa}
  A single column $\Row\times\{c\}$, a single row $\{r\}\times\Col$,
  the diagonal set
  ${\{(\text{row}\,n,\allowbreak\text{col}\,n)\}}_{n\in\N}$ if $\Row$
  and $\Col$ are copies of $\N$, are 1-un\-con\-di\-tio\-nal basic sequences
  in all $\Sch^p$. In fact, every column section and every row section
  (this is the terminology of \cite[Def.~4.3]{va69}) is a
  1-un\-con\-di\-tio\-nal basic sequence; note that the length of every path
  in the corresponding graph is at most 2.
\end{exa}

Note that the set $I$ is a (completely) 1-un\-con\-di\-tio\-nal basic
sequence in $\Sch^p$ if and only if the relative Schur multipliers
by signs on $\Sch^p_I$ define (complete) isometries. This yields by
Prop.~\ref{pinf}:

\begin{prp}
  \label{pi}
  Let $I\subseteq\Row\times\Col$ and $0<p\le\infty$. If $I$ is
  a real \klammer{vs.~complex} 1-un\-con\-di\-tio\-nal basic sequence in
  $\Sch^\infty$, then $I$ is also a real \klammer{vs.~complex} completely
  1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
\end{prp}

\begin{exa}
  If $\Row=\Col=\{0,\dots,n-1\}$, $1\le p\le\infty$ and $I=\Row\times
  \Col$, then the complex unconditional constant of the basis of
  elementary matrices in $\Sch^p$ is $n^{|1/2-1/p|}$ and coincides with
  its complete unconditional constant (see \cite[Lemma
  8.1.5]{pi98}.) This is also the real unconditional constant if
  $n=2^k$ is a power of 2 as the norm of Schur
  multiplication by the $k$th tensor power $
  \begin{pmatrix}
    1&1\\
    1&-1
  \end{pmatrix}
  ^{\otimes k}$ (the $k$th Walsh matrix) on $\Sch^p$ is
  $\bigl(2^{|1/2-1/p|}\bigr)^k=n^{|1/2-1/p|}$. Let us now show that if
  $n=3$, the real unconditional constant of the basis of elementary
  matrices in $\Sch^\infty$ is $5/3$ and differs from its complex
  unconditional constant, $\sqrt3$. In fact, because the canonical
  bases of $\ell^2_\Col$ and $\ell^2_\Row$ are symmetric, the norm of
  a Schur multiplier by real signs turns out to equal the norm of one
  of the following three Schur multipliers:
  \begin{equation*}
    \begin{pmatrix}
      1&1&1\\1&1&1\\1&1&1
    \end{pmatrix},~
    \begin{pmatrix}
      1&1&1\\1&1&1\\1&1&-1
    \end{pmatrix}\text{or}~
    \begin{pmatrix}
      -1&1&1\\1&-1&1\\1&1&-1
    \end{pmatrix}.
  \end{equation*}
  The first one has norm 1: it defines the identity. The second one has
  the same norm as the Schur multiplier
  $\begin{pmatrix}1&1\\1&-1\end{pmatrix}$, which is $\sqrt2$, because
  the norm of that multiplier equals the norm of its tensor product by
  $\Id_{\ell_2^2}$, which is $
  \begin{pmatrix}
    1&1&1&1\\
    1&1&1&1\\
    1&1&-1&-1\\
    1&1&-1&-1\\
  \end{pmatrix}
  $.  
  By
  Prop.~\ref{transfer} for $\Gamma=\Z/3\Z$, the third one has the same
  norm as the Fourier multiplier $\varphi=(-1,1,1)$ on
  $\Ell^\infty(G)$, where $G=\{z\in\C:z^3=1\}$: as this multiplier acts by convolution with
  $f=-1+z+z^2$,
  its norm is $\|f\|_{\Ell^1(G)}$, that is
  \begin{equation*}
    \bigl( |-1+1+1| + |-1+\e^{2\iu\pi/3}+\e^{4\iu\pi/3}| +
    |-1+\e^{4\iu\pi/3}+\e^{2\iu\pi/3}| \bigr) /3= 5/3.
  \end{equation*}
  Complex interpolation yields that the real unconditional constant of
  the basis of elementary matrices is in fact strictly less than its
  complex counterpart in all $\Sch^p$ with $p\ne2$.
\end{exa}

\section{\texorpdfstring{\sloppy Varopoulos' characterisation of unconditional matrices in $\Sch^\infty$}{Varopoulos' characterisation of unconditional matrices in the space of bounded operators}}\label{V}\label{ss5}

\noindent Our results may be seen as the isometric counterpart to results by
Varopoulos \cite{va69} on tensor algebras over discrete spaces and
their generalisation to $\Sch^p$. He characterised unconditional basic
sequences of elementary matrices in $\Sch^\infty$ in his study of the
projective tensor product $\cnaught\projtens \cnaught$. We gather up his results
in the next theorem, as they are difficult to extract from the
literature.

\begin{thm}\label{varo}
  Let $I\subseteq\Row\times\Col$. The following are equivalent.
  \begin{enumerate}
  \item $I$ is an unconditional basic sequence in $\Sch^\infty$.
  \item $I$ is an \emph{interpolation set} for Schur multipliers on $\Sch^\infty$:
    every bounded sequence on $I$ is the restriction of a Schur multiplier on $\Sch^\infty$.
  \item $I$ is a \emph{$\Varopoulos$-Sidon set} as defined in
    \cite[Def.~4.1]{va69}: every null
    sequence on $I$ is the restriction of the sequence of coefficients of
    a tensor in $\cnaught(\Col)\projtens \cnaught(\Row)$.
  \item The coefficients of every tensor in
    $\ell^1_\Col\injtens\ell^1_\Row$ with support in
    $I$ form an absolutely convergent series.
  \item $(z_cz'_r)_ {(r,c)\in I}$ is a \emph{Sidon set} in the dual of
    $\T^{\Col}\times\T^{\Row}$, that is, an unconditional basic sequence in
    $\Cont(\T^{\Col}\times\T^{\Row})$.
  \item There is a constant $\lambda$ such that for all
    $\Row'\subseteq\Row$ and $\Col'\subseteq\Col$ with $n$ elements
    $\card[I\cap\Row'\times\Col']\le\lambda n$.
  \item $I$ is a finite union of forests.
  \item $I$ is a finite union of row sections and column sections.
  \item Every bounded sequence supported by $I$ is a Schur
    multiplier on $\Sch^\infty$.
  \end{enumerate}
  \end{thm}

\emph{Sketch of proof. } $(a)\Rightarrow(b)$. If $(a)$ holds, every sequence of signs
$\epsilon\in\{-1,1\}^I$ is a Schur multiplier on $\Sch^\infty_I$. By a
convexity argument, this implies that every bounded sequence is a
Schur multiplier on $\Sch^\infty_I$, which may be extended to a Schur
multiplier on $\Sch^\infty$ with the same norm by
\cite[Cor.~3.3]{pps89}.

$(b)\Rightarrow(c)$ holds by Grothendieck's inequality (see \cite[\S\,5]{pi01}) and an
approximation argument.

$(d)$ is but the formulation dual to $(c)$ (see \cite[\S\,6.2]{va67}.)

$(d)\Rightarrow(e)$. A computation yields
\begin{equation}\label{sidon}
  \biggl\|\sum_{(r,c)\in I}a_{rc}\e_c\otimes\e_r\biggr\|
  _{\ell^1_\Col\injtens\ell^1_\Row} =
  \sup_{|z_c|,|z'_r|=1}\biggl|\sum_{(r,c)\in I}a_{rc}z_cz'_r\biggr|.
\end{equation}

$(e)\Rightarrow(f)$ is \cite[Th.~4.2]{va69}. (The proof can be found in
\cite[\S\,6.3]{va67} and in \cite[\S\,5]{va68}.)

$(f)\Rightarrow(g)$, $(f)\Rightarrow(h)$ can be found in \cite[Th.~6.1]{va68}.

$(g)\Rightarrow(h)$. In fact, a forest is the union of a column section
$I'$ with a row section $I''$ (a \emph{bisection} in the
terminology of \cite[Def.~4.3]{va69}.) It suffices to prove this for a
tree. Let its vertices be indexed by words as described in the
Terminology. Then the set $I'$ of couples of the form $(w,w\concat c)$
with $w$ a word and $c$ a letter is a column section; the set $I''$ of
couples of the form $(w\concat r,w)$ with $w$ a word and $r$ a letter
is a row section.

$(h)\Rightarrow(i)$ is \cite[Th.~4.5]{va68}. Note that row sections
and column sections form 1-un\-con\-di\-tio\-nal basic sequences in
$\Sch^\infty$ and are 1-complemented in $\Sch^\infty$ by Prop.~\ref{complem}.

$(i)\Rightarrow(a)$ follows from the open mapping theorem.

\section{Closed walk relations}
\label{ss6}

\noindent We now introduce and study the combinatorial objects that we need in
order to analyse the expansion of the function defined by
\begin{equation}\label{Phidef}
  \Phi_I(\epsilon,a)= 
  \tr\Bigl|\sum_{q\in I}\epsilon_qa_q\e_q\Bigr|^p
\end{equation}
for $I\subseteq\Row\times\Col$, a positive even integer $p=2k$, signs $\epsilon_q\in\T$ and
coefficients $a_q\in\C$, of which only a finite number are nonzero. In fact,
\begin{equation}
  \label{compu}
\begin{split}
  \Phi_I(\epsilon,a)&=\tr\Bigl(\sum_{(r,c),(r',c')\in I}
  (\epsilon_{rc}a_{rc}\e_{rc})^*
  (\epsilon_{r'c'}a_{r'c'}\e_{r'c'})
  \Bigr)^k\\
  &=\tr~\sum_{\substack{(r_1,c_1),(r'_1,c'_1),\dots,\\
      (r_k,c_k),(r_k',c'_k)\in I}}
  ~\prod_{i=1}^k
  (\epsilon_{r_ic_i}^{-1}\overline{a_{r_ic_i}}\e_{c_ir_i})
  (\epsilon_{r'_ic'_i}a_{r'_ic'_i}\e_{r'_ic'_i})\\
  &=\sum_{\substack{(r_1,c_1),(r_1,c_2),\dots,\\
      (r_k,c_k),(r_k,c_{k+1})\in I}}
  ~\prod_{i=1}^k\epsilon_{r_ic_i}^{-1}\epsilon_{r_ic_{i+1}}
  \overline{a_{r_ic_i}}a_{r_ic_{i+1}}\quad(\text{where}~c_{k+1}=c_1.)
\end{split}
\end{equation}
The latter sum runs over all closed walks
$(c_1,r_1,c_2,\dots,c_k,r_k)$ of length $p$ in the graph $I$. With
multinomial notation, its terms have the form
$\epsilon^{\beta-\alpha}\overline{a}^\alpha a^\beta$ with
$|\alpha|=|\beta|=k$. The attempt to describe those couples
$(\alpha,\beta)$ that effectively arise in this expansion yields the
following definition.

\begin{dfn}\label{cwd}
  Let $p=2k\ge0$ be an even integer and $I\subseteq\Row\times\Col$. 

  \begin{enumerate}
  \item Let $\Alpha_k^I=\{\alpha\in\N^I:\sum_{q\in
      I}\alpha_q=k\}$ and set
    \begin{equation*}
      \textstyle
      \Beta_{k}^I=
      \bigl\{(\alpha,\beta)\in\Alpha_k^I\times\Alpha_k^I:
      \forall\, r\,\sum_c\alpha_{rc}=\sum_c\beta_{rc}~\text{and}~
      \forall\, c\,\sum_r\alpha_{rc}=\sum_r\beta_{rc}\bigr\}.
    \end{equation*}
  \item Two couples $(\alpha^1,\beta^1)\in\Beta_{k_1}^I$,
    $(\alpha^2,\beta^2)\in\Beta_{k_2}^I$ are \emph{disjoint} if
    $k_1,k_2\ge1$ and 
    \begin{equation}\label{alpha}
      \alpha^1_{rc}\ge1 \quad\Rightarrow\quad \forall\,(r',c)\in
      I\quad \alpha^2_{r'c}=0 \quad\text{and}\quad \forall\,(r,c')\in
      I\quad \alpha^2_{rc'}=0.
    \end{equation}
  \item The set $\WR_{k}^I$ of \emph{closed walk relations} of length $p$ in $I$ is
    the subset of those $(\alpha,\beta)\in\Beta_{k}^I$ that cannot be
    decomposed into the sum of two disjoint couples.
  \item Let $\Walk_k^I$ be the set of closed
    walks of length $p$ in the graph $I$. To every closed walk 
    $P=(c_1,r_1,c_2,r_2,\dots,c_k,r_k)$ of length $p$ we associate the
    couple $(\alpha,\beta)\in\Alpha_k^I\times\Alpha_k^I$ defined by 
    \begin{align*}
      \alpha_q&=\card\bigl[i\in\{1,\dots,k\}:(r_i,c_i)=q\bigr]\\
      \beta_q&=\card\bigl[i\in\{1,\dots,k\}:(r_i,c_{i+1})=q\bigr]
      \quad(\text{where}~c_{k+1}=c_1.)
    \end{align*}
    We shall write $P\sim (\alpha,\beta)$ and call $n_{\alpha\beta}$
    the number of elements of $\Walk_k^I$ mapped onto
    $(\alpha,\beta)$.
  \end{enumerate}
  
\end{dfn}

Note that the conditions in Eq.~\eqref{alpha} is in fact symmetric and
that it may be stated with $\beta^1$ and $\beta^2$ instead of $\alpha^1$
and $\alpha^2$.

\begin{exa}
  Let $\Row=\Col=\{0,1,2,3\}$ and $I=\Row\times\Col$. The couple
  $(\e_{00}+\allowbreak\e_{11}+\allowbreak\e_{22}+\allowbreak\e_{33},
  \allowbreak\e_{01}+\allowbreak\e_{10}+\allowbreak\e_{23}+\allowbreak\e_{32})$
  is an element of $\Beta_{4}^I\setminus\WR_{4}^I$: it is the sum of the
  two disjoint closed walk relations
  $(\e_{00}+\allowbreak\e_{11},\allowbreak\e_{01}+\allowbreak\e_{10})$
  and
  $\allowbreak(\e_{22}+\allowbreak\e_{33},\allowbreak\e_{23}+\allowbreak\e_{32})$.
\end{exa}

\begin{exa} Let $I=\Row\times\Col=\{0,1\}\times\{0,1\}$. Two closed
  walks are associated with the closed walk relation
  $(\e_{00}+\allowbreak \e_{11},\allowbreak \e_{01}+\allowbreak
  \e_{10})\in\WR_{2}^I$: the two cycles $(\text{col}\,0,\allowbreak
  \text{row}\,0,\allowbreak \text{col}\,1,\allowbreak \text{row}\,1)$
  and $(\text{col}\,1,\allowbreak \text{row}\,1,\allowbreak
  \text{col}\,0,\allowbreak \text{row}\,0)$.  Six closed walks are
  mapped onto the closed walk relation $(2\e_{00}+2\e_{01},\allowbreak
  2\e_{00}+2\e_{01})$: the $\dfrac{4!}{2!\,2!}$ concatenations of a
  permutation of $(\text{col}\,1,\allowbreak\text{row}\,0)$,
  $(\text{col}\,1,\allowbreak\text{row}\,0)$,
  $(\text{col}\,0,\allowbreak\text{row}\,0)$,
  $(\text{col}\,0,\allowbreak\text{row}\,0)$.
\end{exa}

The next proposition shows that, for our purpose, closed walk
relations describe entirely closed walks.

\begin{prp}
  \label{gon:bijection}
  Let $p=2k\ge0$ be an even integer and
  $I\subseteq\Row\times\Col$. The image of the mapping in
  Def.~\ref{cwd}$(d)$ is $\WR_{k}^I$:

  \begin{enumerate}
  \item if $P\in\Walk_k^I$ and $P\sim(\alpha,\beta)$, then
    $(\alpha,\beta)\in\WR_{k}^I$;
  \item if $(\alpha,\beta)\in\WR_{k}^I$, then there is a $P\in\Walk_k^I$ such that $P\sim(\alpha,\beta)$, so that
    $n_{\alpha\beta}\ge1$.
  \end{enumerate}

\end{prp}

\begin{proof} 
  $(a)$. Let $P=(c_1,r_1,c_2,r_2,\dots,c_k,r_k)$. In fact, 
  \begin{gather*}
    \sum\nolimits_c\alpha_{rc}=\card[i\in\{1,\dots,k\}:r_i=r]=\sum\nolimits_c\beta_{rc}\\
    \sum\nolimits_r\alpha_{rc}=\card[i\in\{1,\dots,k\}:c_i=c]=\sum\nolimits_r\beta_{rc}
  \end{gather*}
  and $(\alpha,\beta)\in\Beta_{k}^I$. If
  $(\alpha,\beta)=(\alpha^1,\beta^1)+(\alpha^2,\beta^2)$ with 
  $(\alpha^i,\beta^i)\in\Beta_{k_i}^I$ and $k_i\ge1$, there is an $i$ such
  that $\alpha^1_{r_ic_i}\ge1$ and $\alpha^2_{r_{i+1}c_{i+1}}\ge1$
  (where $(r_{i+1},c_{i+1})=(r_1,c_1)$ if $i=k$.) If
  $\beta^1_{r_ic_{i+1}}\ge1$, then $\sum_r\alpha^1_{rc_{i+1}}\ge1$, so
  that there is an $r$ such that
  $\alpha^1_{rc_{i+1}}\ge1$. Otherwise $\beta^2_{r_ic_{i+1}}\ge1$, so
  that $\sum_c\alpha^2_{r_ic}\ge1$ and there is a $c$ such that
  $\alpha^2_{r_ic}\ge1$. Therefore $(\alpha^1,\beta^1)$ and
  $(\alpha^2,\beta^2)$ are not disjoint and $(\alpha,\beta)\in\WR_{k}^I$.

  $(b)$. We have to find a closed walk of length $p$ that is mapped onto
  $(\alpha,\beta)$. If $k=0$, the
  empty closed walk suits.  Suppose that $k\ge1$; Consider a walk
  $(c_1,r_1,c_2,r_2,\dots,c_j,r_j,c_{j+1})$ in $I$ such that
  $\alpha^1_q=\card[i:(r_i,c_i)=q]\le\alpha_q$ and
  $\beta^1_q=\card[i:(r_i,c_{i+1})=q]\le\beta_q$ for every
  $q\in\Row\times\Col$, and furthermore $j$ is maximal.  We claim $(A)$
  that $c_{j+1}=c_1$ and $(B)$ that $j=k$. Let
  $(\alpha^2,\beta^2)=(\alpha,\beta)-(\alpha^1,\beta^1)$.

  $(A)$. If $c_{j+1}\ne c_1$, then
  \begin{gather*}
    \sum\nolimits_r\alpha^1_{rc_{j+1}}=\card[i\in\{1,\dots,j\}:c_i=c_{j+1}]\\
    \sum\nolimits_r\beta^1_{rc_{j+1}}=\card[i\in\{1,\dots,j+1\}:c_i=c_{j+1}] =
    1 + \sum\nolimits_r\alpha^1_{rc_{j+1}},
  \end{gather*}
  so that there must be $r_{j+1}$ with
  $\alpha^2_{r_{j+1}c_{j+1}}\ge1$. But then
  \[\sum\nolimits_c\beta^2_{r_{j+1}c}=\sum\nolimits_c\alpha^2_{r_{j+1}c}\ge1\] and there
  must be $c_{j+2}$ such that $\beta^2_{r_{j+1}c_{j+2}}\ge1$: $j$ is not
  maximal.

  $(B)$. Suppose that $j<k$. Then $(\alpha^1,\beta^1)\in\Beta_{j}^I$ and  
  $(\alpha^2,\beta^2)\in\Beta_{k-j}^I$. By hypothesis, they are not disjoint: there are $r,c,c'$ 
  such that $\alpha^1_{rc}\alpha^2_{rc'}\ge1$ or $r,r',c$ such
  that $\alpha^1_{rc}\alpha^2_{r'c}\ge1$. By interchanging $\Row$ and
  $\Col$ and by relabelling the vertices if necessary, we may suppose without loss of
  generality that for $r'_1=r_j$ there is $c'_1$ such that
  $\alpha^2_{r'_1c'_1}\ge1$. Then there is $c'_2$ such that
  $\beta^2_{r'_1c'_2}\ge1$. By the argument used in Claim $(A)$, there is a
  closed walk 
  $(c'_1,r'_1,c'_2,\dots,c'_{j'},r'_{j'})$ such that
  $\card[i:(r'_i,c'_i)=q]\le\alpha^2_q$ and
  $\card[i:(r'_i,c'_{i+1})=q]\le\beta^2_q$ (where $c'_{j'+1}=c'_1$.) 
  Then the closed walk 
  \begin{equation*}(c_1,r_1,c_2,r_2,\dots,c_j,r_j,c'_2,r'_2,
    \dots,c'_{j'},r'_{j'},c'_1,r'_1)\end{equation*}
  shows that $j$ is not maximal.  
\end{proof}

We are now in position to state the following theorem, a matrix counterpart to the computation
presented in \cite[Prop.~2.5$(ii)$]{ne98}.

\begin{thm}\label{sigma:ldc}
  Let $p=2k$ be a positive even integer and $I\subseteq\Row\times\Col$. 

  \begin{enumerate}
  \item The function $\Phi_I$ in Eq.~\eqref{Phidef} has the expansion
    \begin{equation}\label{gon:expand}
      \Phi_I(\epsilon,a) = 
      \sum_{(\alpha,\beta)\in\WR_{k}^I}
      n_{\alpha\beta}\epsilon^{\beta-\alpha}\overline{a}^\alpha
      a^\beta,
    \end{equation}
    where $n_{\alpha\beta}\ge1$ for every 
    $(\alpha,\beta)\in\WR_{k}^I$.
  \item If $\epsilon\in\T^I$ and $a\in(\Sch^p)^I$ is finitely
  supported, then the function
    \begin{equation}\label{gon:psi}
      \Psi_I(\epsilon,a) = \tr\Bigl|\sum_{q\in
          I}\epsilon_qa_q\e_q\Bigr|^p
    \end{equation}
    has the expansion
    \begin{equation}\label{gon:expand2}
    \sum_{(\alpha,\beta)\in\WR_{k}^I}
      \epsilon^{\beta-\alpha}\sum_{(c_1,r_1,\dots,c_k,r_k)\sim(\alpha,\beta)}
      \prod_{i=1}^ka_{r_ic_i}^*a_{r_ic_{i+1}}~(\text{with}~c_{k+1}=c_1.)
   \end{equation}
  \end{enumerate}

\end{thm}

\begin{proof} 
  This follows from Def.~\ref{cwd} and Prop.~\ref{gon:bijection}.
\end{proof}

Note that the edges of a closed walk $P\sim(\alpha,\beta)$
are precisely those $\{r,c\}$ such that $\alpha_{rc}+\beta_{rc}\ge1$.
$P$ is a cycle if and only if $P$ does not have length 0 or 2 and
$\sum_r\alpha_{rc}\le1$ for all $c$ and
$\sum_c\alpha_{rc}\le1$ for all $r$. We now show how to decompose
closed walks into cycles.

\begin{prp}\label{gon:dec}
  Let $P=(c_1,r_1,c_2,r_2,\dots,c_k,r_k)\sim(\alpha,\beta)$ be a closed walk.
  
  \begin{enumerate}
  \item If $r_i=r_j$ \klammer{vs.~$c_i=c_j$} for some
    $i\ne j$, then $P$ is the juxtaposition of two nonempty closed walks
    $P_1\sim(\alpha^1,\beta^1)$ and $P_2\sim(\alpha^2,\beta^2)$ such
    that $(\alpha,\beta)=(\alpha^1,\beta^1)+(\alpha^2,\beta^2)$ and
    $\sum_c\alpha^1_{r_ic}\,,\,\sum_c\alpha^2_{r_ic}\ge1$
    \klammer{vs.~$\sum_r\alpha^1_{rc_i}\,,\,\sum_r\alpha^2_{rc_i}\ge1$.}
  \item $P$ is the juxtaposition of nonempty closed walks
    $P_j\sim(\alpha^j,\beta^j)$ such that
    $\sum_r\alpha^j_{rc}\le1$ for all $c$,
    $\sum_c\alpha^j_{rc}\le1$ for all $r$ and 
    $(\alpha,\beta)=\sum(\alpha^j,\beta^j)$.
  \item There are cycles $P_j\sim(\alpha^j,\beta^j)$ and a $\gamma$ such that
    $(\alpha,\beta)=(\gamma,\gamma)+\sum(\alpha^j,\beta^j)$.
  \end{enumerate}
  
\end{prp}

\begin{proof}
  $(a)$. If $r_i=r_j$ for $i<j$, we may suppose that $j=k$: consider
  the closed walks 
  $P_1=(c_1,\allowbreak r_1,\dots,\allowbreak c_i,\allowbreak r_i)$ and
  $P_2=(c_{i+1},r_{i+1},\dots,c_k,r_k)$. If $c_i=c_j$ for $i<j$, we
  may suppose that $i=1$: consider then
  $P_1=(c_1,r_1,\dots,c_{j-1},r_{j-1})$ and
  $P_2=(c_j,r_j,\dots,c_k,r_k)$.
  
  $(b)$. Use $(a)$ in a maximality argument.
  
  $(c)$. Note that the closed walks $P_j$ in $(b)$ are either cycles
  or have length 2; in the latter case $P_j=q\sim(\e_q,\e_q)$ for some
  $q\in I$.
\end{proof}

\section{Schur multipliers on a cycle}\label{ss7}
\label{sec:cst}

\noindent We can realise a cycle of even length $2s$, $s\ge2$, in the following
convenient way. Let $\Gamma=\Z/s\Z$. Then the adjacency relation of
integers modulo $s$ turns $\Gamma$ into the cycle $(0,1,\dots,s-1)$ of
length $s$. We double this cycle into the bipartite cycle
$(\text{col}\,0,\allowbreak \text{row}\,0,\allowbreak
\text{col}\,1,\allowbreak \text{row}\,1,\dots,\allowbreak
\text{col}\,{s-1},\allowbreak \text{row}\,{s-1})$ on $\Gamma\amalg
\Gamma$, corresponding to the set of couples $I=\{(i,i),(i,i+1):i\in
\Gamma\}\subseteq \Gamma\times \Gamma$: $I$ is the pattern
\[
\bordermatrix{
  &\scriptstyle0&\scriptstyle1&\scriptstyle2&\scriptstyle\cdots&\scriptstyle s-2&\scriptstyle s-1\cr
  \hfil\scriptstyle0&*&*&0&\ddots&0&0\cr
  \hfil\scriptstyle1&0&*&*&\ddots&0&0\cr
  \hfil\scriptstyle2&0&0&*&\ddots&0&0\cr
  \hfil\scriptstyle\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\cr
  \hfil\scriptstyle s-2&0&0&0&\ddots&*&*\cr
  \hfil\scriptstyle s-1&*&0&0&\ddots&0&*\cr}\text.
\]
$\Gamma$ is the group dual to $G=\hat \Gamma=\{z\in\C:z^s=1\}$.  We
shall consider the space $\Ell^p_\Lambda(G)$ spanned by
$\Lambda=\{1,z\}$ in $\Ell^p(G)$, where $z$ is the identical function
on $G$: its norm is given by
$\|a+bz\|_{\Ell^p(G)}=\big(s^{-1}\sum_{z^s=1}|a+bz|^p\bigr)^{1/p}$.

\begin{prp}
  \label{cyno}
  Let $0<p\le\infty$, $s\ge2$ and $I=\{(i,i),(i,i+1):i\in\Z/s\Z\}$.
  Let $\epsilon\in\T^I$ be a Schur multiplier by signs on
  $\Sch^p_I$.
  
  \begin{enumerate}
  \item The Schur multiplier $\epsilon$ has the same norm as the Schur multiplier
    $\hat\epsilon$ given by $\hat\epsilon_q=1$ for
    $q\ne({s-1},0)$ and $\hat\epsilon_{s-1,0} =
    \overline{\epsilon_{00}} \epsilon_{01} \dots
    \overline{\epsilon_{{s-1},{s-1}}}
    \epsilon_{{s-1},0}$.
  \item The Schur multiplier $\epsilon$ has the same norm as
    $\check\epsilon$ given by $\check\epsilon_{ii} = 1$ and
    $\check\epsilon_{i,i+1}=\vartheta$ with $\vartheta$ any
    $s$th root of $\hat\epsilon_{{s-1},0}$ or its complex
    conjugate: without loss of generality, $\vartheta=\e^{\iu\alpha}$
    with $\alpha\in[0,\pi/s]$.
  \item The norm of $\epsilon$ on $\Sch^p_I$ is bounded below by the
    norm of the relative Fourier multiplier $\mu:a+bz\mapsto
    a+\vartheta bz$ on $\Ell^p_\Lambda(G)$; their complete norms are equal.
  \item The norm of $\epsilon$ on $\Sch^1_I$ and on
    $\Sch^\infty_I$ is equal to the norm of $\mu$ on
    $\Ell^1_\Lambda(G)$ and on $\Ell^\infty_\Lambda(G)$: this norm is
    \begin{equation*}
      \displaystyle\frac{\cos(\alpha/2-\pi/2s)}{\cos\pi/2s}
      =\frac{\max_{z^s=-1}|\vartheta+z|}{|1+\e^{\iu\pi/s}|}\text.
    \end{equation*}
  \item The Schur multiplication operator $\Schur_\epsilon$ is an
    isometry on $\Sch^p_I$ if and only if
    $p/2\in\{1,2,\dots,s-1\}$ or $\overline{\epsilon_{00}}
    \epsilon_{01} \dots \overline{\epsilon_{{s-1},{s-1}}}
    \epsilon_{{s-1},0}=1$.
  \end{enumerate}
  
\end{prp}

\begin{proof}
  $(a)$ and $(b)$ follow from the matrix unconditionality of
  Schatten-von-Neu\-mann norms (see Eq.~\eqref{mu}) and from the fact
  that the Schur multipliers $\epsilon$ and
  $\bar\epsilon=(\overline{\epsilon_q})_{q\in I}$ have the same norm
  on $\Sch^p_I$.
  
  $(c)$ follows from Prop.~\ref{transfer}.

  $(d)$.  Let us compute
  $f(\beta)={\|1+\e^{\iu\beta}z\|}_{\Ell^1(G)}$. As
  $f(\beta)=f(\beta+2\pi/s)=f(-\beta)$, we may suppose without loss of
  generality that $\beta\in[0,\pi/s]$. Then
  $\lvert\beta/2+k\pi/s\rvert\le\pi/2$ if $-\lfloor s/2\rfloor\le
  k\le\lceil s/2\rceil-1$, so that
  \begin{align*}
    f(\beta)
    &=\frac1s\sum_{k=-\lfloor s/2\rfloor}^{\lceil s/2\rceil-1}
    \bigl\lvert1+\e^{\iu\beta}\e^{2\iu k\pi/s}\bigr\rvert\\
    &=\frac2s\sum_{k=-\lfloor s/2\rfloor}^{\lceil s/2\rceil-1}
    \cos(\beta/2+k\pi/s)\\
    &=\frac2s\Re\biggl(\e^{\iu\beta/2}\frac{\e^{\iu\lceil s/2\rceil\pi/s}-\e^{-\iu\lfloor s/2\rfloor/s}}{\e^{\iu\pi/s}-1}\biggr)\\
    &=\frac2{s\sin(\pi/2s)}\cdot
    \begin{cases}
      \cos(\beta/2-\pi/2s)&\text{if $s$ is even}\\
      \cos(\beta/2)&\text{if $s$ is odd.}
    \end{cases}
  \end{align*}
  This shows in both cases that the norm of $\mu$ on $\Ell^1_\Lambda(G)$ is bounded
  below by $\cos(\alpha/2-\pi/2s)\slash\cos(\pi/2s)$. 
  The complete norm of $\mu$ on $\Ell^\infty_\Lambda(G)$ is equal to
  its norm and thus to the maximum of
  \begin{math}
    g(w)
    ={\|w+\vartheta z\|}_{\Ell^\infty(G)}\slash{\|w+z\|}_{\Ell^\infty(G)}
  \end{math}
  for $w\in\C$. Let $w=r\e^{\iu\beta}$ with $r\ge0$ and $\beta\in\R$.
  Note that
  \begin{equation*}
    {\|w+z\|}_{\Ell^\infty(G)}=\bigl\lvert r+\e^{\iu\mathrm{d}(\beta,(2\pi/s)\Z)}\bigr\rvert
  \end{equation*}
  is a decreasing function of $\mathrm{d}(\beta,(2\pi/s)\Z)$ and that
  \begin{equation*}
    \mathrm{d}(\alpha-\beta,(2\pi/s)\Z)<\mathrm{d}(\beta,(2\pi/s)\Z)
    \ \Leftrightarrow\ 
    \beta\in\mathopen]\alpha/2,\pi/s+\alpha/2\mathclose[\bmod{2\pi/s}\text.
  \end{equation*}
  As
  $g(w)=g(wz)$ if $z^s=1$, we may suppose without loss of
  generality that $\beta\in\mathopen]\alpha/2,\pi/s+\alpha/2\mathclose[$. Therefore
  \begin{equation*}
    g(w)=
    \begin{cases}
      \Bigl|\dfrac{w+\e^{\iu\alpha}}{w+1}\Bigr|&\text{if $\beta\in\mathopen]\alpha/2,\pi/s]$}\\
      \Bigl|\dfrac{w+\e^{\iu\alpha}}{w+\e^{2\iu\pi/s}}\Bigr|&\text{if
        $\beta\in[\pi/s,\pi/s+\alpha/2\mathclose[$.}
    \end{cases}
  \end{equation*}
  As $g$ tends to 1 at infinity and $g(w)=1$ if
  $\beta\in\{\alpha/2,\pi/s+\alpha/2\}$, the maximum principle shows
  that $g$ attains its maximum with $\beta=\pi/s$. Finally,
  \begin{align*}
    {g(r\e^{\iu\pi/s})}^2
    &=\frac{1+2r\cos(\pi/s-\alpha)+r^2}{1+2r\cos(\pi/s)+r^2}\\
    &=1+\frac{\cos(\pi/s-\alpha)-\cos\pi/s}{\cos(\pi/s)+(r+1/r)/2}
    \le{g(\e^{\iu\pi/s})}^2
    ={\biggl(\frac{\cos(\pi/2s-\alpha/2)}{\cos\pi/2s}\biggr)}^2\text.
  \end{align*}
    $(e)$. If $p$ is not an even integer and $\vartheta^s\ne1$, then
  $\mu$ is not an isometry on $\Ell^p_\Lambda(G)$: otherwise the
  functions $z$ and $\vartheta z$ would have the same distribution by
  the Plotkin-Rudin Equimeasurability Theorem (see \cite[Th.~2]{kk01}). If
  $p\in\{2,4,\dots,2s-2\}$, then $I$ contains no cycle of length
  $4,6,\dots,p$, so that by Prop.~\ref{gon:dec}$(c)$ every closed walk
  $P\sim(\alpha,\beta)$ satisfies $\alpha=\beta$. The function
  $\Phi_I(\epsilon,a)$ in Eq.~\eqref{Phidef} is therefore constant in
  $\epsilon$ by Th.~\ref{sigma:ldc}$(a)$. If $p\in\{2s,2s+2,\dots\}$,
  the closed walk relation
  \begin{equation*}
    (\alpha,\beta) = \Bigl(\sum_{i\in \Gamma}\e_{ii}, \sum_{i\in
      \Gamma}\e_{i,i+1}\Bigr)+(p/2-s)(\e_{00},\e_{00})
  \end{equation*}
  satisfies $n_{\alpha\beta}\ge1$ by Prop.~\ref{gon:bijection}. Then
  the coefficient of $\Phi_I(\epsilon,a)$ in ${\bar a}^\alpha a^\beta$
  equals
  \begin{equation*}
    n_{\alpha\beta}\overline{\epsilon_{00}} \epsilon_{01} \dots
  \overline{\epsilon_{{s-1},{s-1}}}\epsilon_{{s-1},0}
\end{equation*}
  and must equal
  the same quantity with $\epsilon$ replaced by 1 if $\epsilon$
  defines an isometry on $\Sch^p_I$.
\end{proof}

\begin{rem} 
  See \cite[p.~245]{li96} for a similar application of the
  Plotkin-Rudin Equimeasurability Theorem in $(e)$.
\end{rem}

The real unconditional constant of $I$ is therefore the norm of
$\check\epsilon$ with $\alpha=\pi/s$, and the complex
unconditional constant is the maximum of the norm of $\check\epsilon$ for
$\alpha\in[0,\pi/s]$.
This yields

\begin{cor}\label{cyc}
  Let $0<p\le\infty$ and $s\ge2$. Let $I$ be the cycle of
  length $2s$.

  \begin{enumerate}
  \item $I$ is a real 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$
    if and only if $p\in\{2,\allowbreak 4,\dots,\allowbreak 2s-2\}$. 
  \item The real and complex unconditional constants of $I$
    in the spaces $\Sch^1$ and $\Sch^\infty$ equal $\sec\pi/2s$.
  \end{enumerate}

\end{cor}

\section{\texorpdfstring{1-unconditional matrices in $\Sch^p$,  $p$ not an even integer}{1-unconditional matrices in p-Schatten classes,  p not an even integer}}\label{ss8}

\noindent We now state the announced isometric counterpart to Varopoulos'
characterisation of unconditional matrices in $\Sch^\infty$ (Section \ref{V})
and its generalisation to $\Sch^p$ for $p$ not an even integer.

\begin{thm}
  \label{p-np}
  Let $I\subseteq\Row\times\Col$ be nonempty and
  $p\in(0,\infty]\setminus2\N$.  The following are
  equivalent.
  
  \begin{enumerate}
  \item $I$ is a complex completely 1-un\-con\-di\-tio\-nal basic
    sequence in $\Sch^p$.
  \item $I$ is a complex 1-un\-con\-di\-tio\-nal basic sequence in
    $\Sch^p$.
  \item $I$ is a real 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item $I$ is a forest.
  \item For each $\epsilon\in\T^I$ there are
    $\zeta\in\T^\Col$  and $\eta\in\T^\Row$ such that
    $\epsilon_{rc}=\zeta(c)\eta(r)$ for all $(r,c)\in I$.
  \item For each $\epsilon\in\{-1,1\}^I$ there are
    $\zeta\in\{-1,1\}^\Col$  and $\eta\in\{-1,1\}^\Row$ such that
    $\epsilon_{rc}=\zeta(c)\eta(r)$ for all $(r,c)\in I$.
  \item $I$ is a \emph{set of $\Varopoulos$-interpolation of constant 1}: for all
    $\varphi\in\ell^\infty_I$ 
    \begin{equation}
      \label{interpol}
      \inf\Bigl\{\biggl\|
      \sum_{(r,c)\in\Row\times\Col}
      \tilde{\varphi}_{rc}\e_c\otimes\e_r
      \biggr\|_{\ell^\infty_\Col\projtens \ell^\infty_\Row}:
      \tilde{\varphi}|_I=\varphi\Bigr\}=\sup_{q\in I}|\varphi_q|.
    \end{equation}
  \item $I$ is a \emph{$\Varopoulos$-Sidon set of constant 1}: for all
    $\varphi\in \cnaught(I)$ 
    \begin{equation}
      \label{sid}
      \inf\Bigl\{\biggl\|
      \sum_{(r,c)\in\Row\times\Col}
      \tilde{\varphi}_{rc}\e_c\otimes\e_r
      \biggr\|_{\cnaught(\Col)\projtens \cnaught(\Row)}:
      \tilde{\varphi}|_I=\varphi\Bigr\}=\sup_{q\in I}|\varphi_q|.
    \end{equation}
  \item For every tensor $u=\sum_{(r,c)\in
      I}a_{rc}\e_c\otimes\e_r$ in
    $\ell^1_\Col\injtens\ell^1_\Row$ with support in $I$
    we have
    $\|u\|_{\ell^1_\Col\injtens\ell^1_\Row}=\sum_{(r,c)\in
      I}|a_{rc}|$.
  \item $(z_cz'_r)_{(r,c)\in I}$ is a \emph{Sidon set of constant 1} in the
    dual of $\T^{\Col}\times\T^{\Row}$, that is, a 1-un\-con\-di\-tio\-nal basic
    sequence in $\Cont(\T^{\Col}\times\T^{\Row})$: if $(a_{rc})$ is
    finitely supported,
    \[\sup_{(z,z')\in\T^\Col\times\T^\Row}
    \Bigl|\sum_{(r,c)\in I}a_{rc}z_cz'_r\Bigr| = \sum_{(r,c)\in
    I}|a_{rc}|\ .\]
  \item For all $\Row'\subseteq\Row$ and $\Col'\subseteq\Col$ with
    $k\ge1$ elements $\card[I\cap\Row'\times\Col']\le2k-1$.
  \item $I$ is an \emph{isometric interpolation set} for Schur multipliers on
    $\Sch^\infty$: every $\varphi\in\ell^\infty_I$ is the restriction of a
    Schur multiplier on $\Sch^\infty$ with norm
    $\|\Schur_\varphi\|=\|\varphi\|_{\ell^\infty_I}$.
  \end{enumerate}

\end{thm}

\begin{proof} 
  $(a)\Rightarrow(b)\Rightarrow(c)$ is trivial.

  $(c)\Rightarrow(d)$. Suppose that $I$ contains a cycle
  $(c_0,r_0,\dots,c_{s-1},r_{s-1})$ with
  $s\ge2$. Cor.~\ref{cyc}$(a)$ shows
  that $I$ is not a real 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.


  $(d)\Leftrightarrow(k)$. A tree on $2k$ vertices has exactly $2k-1$ edges, so
  that a forest $I$ satisfies $(k)$. Conversely, a cycle of length $2s$
  is a graph with $s$ row vertices, $s$ column vertices and $2s$
  edges.
  
  $(d)\Rightarrow(e)$. Suppose first that $I$ is a tree and index the vertices of its
  edges by words $w\in W$ as described in the Terminology. Let us
  define $\eta$ and $\zeta$ inductively. If $r$ is the root
  of the tree, indexed by $\emptyset$, let $\eta(r)=1$. Suppose that $\eta$ and
  $\zeta$ have been defined for all vertices indexed by words of
  length at most $2n$. If $c$ is indexed by a word $w$ of length
  $2n+1$, let $r$ be the vertex indexed by the word of length $2n$
  with which $w$ begins and let $\zeta(c)=\epsilon(r,c)\slash\eta(r)$. If $r$ is
  indexed by a word $w$ of length $2n+2$, let $c$ be the vertex
  indexed by the word of length $2n+1$ with which $w$ begins and let
  $\eta(r)=\epsilon(r,c)\slash\zeta(c)$.  If $I$ is a union of pairwise
  disjoint trees, we may define $\eta$ and $\zeta$ on each tree
  separately. We may finally extend $\eta$ to $\Row$ and $\zeta$ to
  $\Col$ in an arbitrary manner.
    
  $(d)\Rightarrow(f)$ may be proved as $(d)\Rightarrow(e)$.
  
  $(f)\Rightarrow(c)$. If $(f)$ holds, then every Schur multiplier by
  signs $\epsilon\in\{-1,1\}^I$ is elementary in the sense that
  $\epsilon=\zeta\otimes\eta$. The complete norm of $\Schur_\epsilon$ on
  any $\Sch^p_I$ is therefore
  $\|\zeta\|_{\ell^\infty_\Col}\|\eta\|_{\ell^\infty_\Row}=1$.
  
  $(e)\Rightarrow(g)$. If $(e)$ holds, every
  $\varphi\in\T^I\subseteq\ell^\infty_I$ may be extended to an
  elementary tensor $\zeta\otimes\eta$ of norm 1. $(g)$ follows
  because every element of $\ell^\infty_I$ with norm 1 is the half sum
  of two elements of $\T^I$: note that $\e^{\iu t}\cos u =
  \bigl(\e^{\iu(t+u)}+\e^{\iu(t-u)}\bigr)/2$.

  $(g)\Rightarrow(h)$. It suffices to check Equality~\eqref{sid} for $\varphi$ with support
  contained in a finite rectangle set $\Row'\times\Col'$. As $\ell^\infty_{\Col'}\projtens \ell^\infty_{\Row'}$
  is a subspace of $\ell^\infty_\Col\projtens \ell^\infty_\Row$,
  Eq.~\eqref{interpol} yields Eq.~\eqref{sid}.

  $(h)\Leftrightarrow(i)$ because they are dual statements.

  $(i)\Leftrightarrow(j)$. Use Equality \eqref{sidon}.

  $(h)\Rightarrow(l)$ may be deduced by the argument of
  Prop.~\ref{complem}$(a)\Rightarrow(b)$.

  $(l)\Rightarrow(a)$. Taking sign sequences $\varphi\in\T^I$ in $(l)$ shows that
  all relative Schur multipliers by signs on
  $\Sch^\infty_I$ define isometries. Apply Prop.~\ref{pi}. \end{proof}

\begin{rem}
  The equivalence of $(e)$ with $(j)$ may also be shown as a consequence of the
  characterisation of Sidon sets of constant 1 in \cite{chm81}.
\end{rem}

Let us now answer Question~\ref{q3}.

\begin{cor}
  Let $I\subseteq\Row\times\Col$. The following are equivalent.

  \begin{enumerate}
  \item For all $\varphi\in \cnaught(I)$ one has $\bigl\|\sum_{(r,c)\in I}
    \varphi_{rc} \e_c \otimes \e_r
    \bigr\|_{\cnaught(\Col)\projtens \cnaught(\Row)}=\sup_{q\in I}|\varphi_q|$.
  \item There are pairwise disjoint sets $\Row_j\subseteq\Row$ and pairwise disjoint sets
    $\Col_j\subseteq\Col$ such that $\Row_j$ or $\Col_j$ is a singleton for each $j$
    and $I=\bigcup\Row_j\times\Col_j$: $I$ is the union of the column
    section $\bigcup_{\card\Row_j=1}\Row_j\times\Col_j$ with the
    disjoint row section $\bigcup_{\card\Row_j>1}\Row_j\times\Col_j$.
  \item $I$ is a union of pairwise disjoint star graphs: every path in $I$ has
    length at most 2.
  \end{enumerate} 
  
\end{cor}

\begin{proof} 
  $(a)\Rightarrow(b)$ follows from
  Prop.~\ref{complem}$(a)\Rightarrow(d)$ and
  Th.~\ref{p-np}$(g)\Rightarrow(d)$.

  $(b)\Leftrightarrow(c)$. $(b)$ holds if and only if $(r,c),(r',c),(r,c')\in
  I\Rightarrow(r=r'~\text{or}~c=c')$ and therefore if and only if $(c)$ holds.

  $(b)\Rightarrow(a)$. Suppose $(b)$ and let $\varphi\in \cnaught(I)$.
  Let $\alpha_j=\sup_{(r,c)\in\Row_j\times\Col_j}|\varphi_{rc}|^{1/2}$. If
  $\alpha_j=0$, let us define $\varrho^j=0$ and $\gamma^j=0$. Otherwise, if
  $\Row_j$ is a singleton $\{r\}$, let us define $\varrho^j=\alpha_j\e_r$ and
  $\gamma^j$ by $\gamma^j_c=\varphi_{rc}/\alpha_j$ if $c\in\Col_j$ and
  $\gamma^j_c=0$ otherwise.  Otherwise, $\Col_j$ is a singleton $\{c\}$ and we
  define $\gamma^j=\alpha_j\e_c$ and $\varrho^j$ by
  $\varrho^j_r=\varphi_{rc}/\alpha_j$ if $r\in\Row_j$ and $\varrho^j_r=0$ otherwise.
  Note that the $\gamma^j$ have pairwise disjoint support and are null sequences,
  as well as the $\varrho^j$. Then
  \begin{equation*}
    \sum_{(r,c)\in I}\varphi_{rc}\e_c\otimes\e_r = 
    \sum_j\gamma^j\otimes\varrho^j = 
    \average_{\epsilon_j=\pm1}
    \biggl(\sum_j\epsilon_j\gamma^j\biggr)
    \otimes\biggl(\sum_j\epsilon_j\varrho^j\biggr)
  \end{equation*}
  is an average of elementary tensors in $\cnaught(\Col)\projtens \cnaught(\Row)$
  of norm $\sup_{q\in I}|\varphi_q|$, so that this average 
  is also 
  bounded by this norm, which obviously is a lower bound. 
\end{proof}

\section{\texorpdfstring{1-unconditional
    matrices in $\Sch^p$, $p$ an even integer}{1-unconditional
    matrices in p-Schatten classes, p an even integer}}
\label{ss9}

\noindent Let us now prove Theorem~\ref{th5} as a consequence of 
Theorem \ref{sigma:ldc} together with Proposition \ref{gon:dec}$(c)$.

\begin{thm}\label{sigma:THM}
  Let $I\subseteq\Row\times\Col$ and $p=2k$ a positive even integer. The following
  assertions are equivalent.
  
  \begin{enumerate}
  \item $I$ is a complex completely 1-un\-con\-di\-tio\-nal basic sequence in
    $\Sch^p$. 
  \item $I$ is a complex 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item For every finite subset $F\subseteq I$ there is an operator $x\in
    \Sch^p$, whose support $S$ contains $F$, such that
    $\bigl\|\sum\epsilon_qx_q\e_q\bigr\|_p$ does not depend on the complex choice
    of signs $\epsilon\in\T^S$.
  \item $I$ is a real
    1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item For every finite subset $F\subseteq I$ there is an operator $x\in
    \Sch^p$ with real matrix coefficients, whose support $S$ contains
    $F$, such that $\bigl\|\sum\epsilon_qx_q\e_q\bigr\|_p$ does not depend on the
    real choice of signs $\epsilon\in\{-1,1\}^S$.
  \item Every closed walk $P\sim(\alpha,\beta)$ of length $2s\le
    2k$ in $I$ satisfies $\alpha=\beta$.
  \item $I$ does not contain any cycle of length $2s\le 2k$ as a subgraph.
  \item For each $v,w\in V$ there is at most one path in $I$ of
    length $l\le k$ that joins $v$ to $w$.
  \end{enumerate}
  
\end{thm}

\begin{proof} 
  $(a)\Rightarrow(b)\Rightarrow(c)$, $(b)\Rightarrow(d)\Rightarrow(e)$ are trivial.
  
  $(c)\Rightarrow(g)$. Suppose that $I$ contains a cycle
  $P\sim(\gamma,\delta)$ of length $2s\le 2k$: the corresponding set
  of couples is $F=\{q:\gamma_q+\delta_q=1\}$. Let $x$ be as in $(c)$
  and let $(\alpha,\beta)=(\gamma,\delta)+(k-s)(\e_q,\e_q)$ for some
  arbitrary $q\in F$.  Then $(\alpha,\beta)\in\WR_{k}^S$. Consider
  $f(\epsilon)=\bigl\|\sum\epsilon_qx_q\e_q\bigr\|_p^p$ as a function
  on the group $\T^S$. Then the Fourier coefficient
  $\widehat{f}(\epsilon^{\beta-\alpha})$ of $f$ at the Steinhaus
  character $\epsilon^{\beta-\alpha}$ is, by Th.~\ref{sigma:ldc}$(a)$,
  \begin{multline*}
    \sum\bigl\{n_{\varepsilon\zeta}\overline{x}^\varepsilon x^\zeta:
    (\varepsilon,\zeta)\in\WR_{k}^S~\text{and}~
    \zeta-\varepsilon=\beta-\alpha\bigr\}\\
    =\overline{x}^\gamma x^\delta
    \sum\bigl\{n_{\varepsilon\zeta}\overline{x}^{\varepsilon-\gamma}
    x^{\zeta-\delta}:
    (\varepsilon,\zeta)\in\WR_{k}^S~\text{and}~
    \zeta-\delta=\varepsilon-\gamma\bigr\}.
  \end{multline*}
  (Note that $\beta-\alpha=\delta-\gamma$.) As this last
  sum has only positive terms and contains at least the term corresponding to
  $(\alpha,\beta)$, $f$ cannot be constant.

  $(e)\Rightarrow(g)$. Let $P\sim(\gamma,\delta)$, $F=\{q:\gamma_q+\delta_q=1\}$ and
  $(\alpha,\beta)$ be as in the proof of the implication $(c)\Rightarrow(h)$. Let $x$ be
  as in $(e)$.  Consider $f(\epsilon)=\bigl\|\sum\epsilon_qx_q\e_q\bigr\|_p^p$ as a
  function on the group $\{-1,1\}^S$. Then the Fourier coefficient
  $\widehat{f}(\epsilon^{\beta-\alpha})$ of $f$ at the
  Walsh character $\epsilon^{\beta-\alpha}$ is, by Th.~\ref{sigma:ldc}$(a)$,
  \begin{multline*}
    \sum\bigl\{n_{\varepsilon\zeta}x^{\varepsilon+\zeta}:
    (\varepsilon,\zeta)\in\WR_{k}^S~\text{and}~
    \zeta-\varepsilon\equiv\beta-\alpha\pmod{2}\bigr\}\\
    =x^{\gamma+\delta}
    \sum\bigl\{n_{\varepsilon\zeta}x^{\varepsilon+\zeta-\gamma-\delta}:
    (\varepsilon,\zeta)\in\WR_{k}^S~\text{and}~
    \zeta-\varepsilon\equiv\delta-\gamma\pmod{2}\bigr\}.
  \end{multline*}
  As this last sum has only positive terms and contains at least the term
  corresponding to $(\alpha,\beta)$, $f$ cannot be constant.

  $(f)\Leftrightarrow(g)$. Apply Prop.~\ref{gon:dec}$(c)$.


  $(g)\Leftrightarrow(h)$. If $I$ contains a cycle $(v_0,\dots,v_{2s-1})$, then $I$
  contains two distinct paths $(v_0,\dots,v_{s})$,
  $(v_0,v_{2s-1},\dots,v_{s})$ of length $s$ from $v_0$ to $v_{s}$. If
  $I$ contains two distinct paths $(v_0,\dots,v_l)$,
  $(v'_0,\dots,v'_{l'})$ with $v_0=v'_0$, $v_{l}=v'_{l'}$ and $l,l'\le
  k$, let $a$ be minimal such that $v_a\ne v'_a$, let $b\ge a$ be minimal such
  that $v_b\in\{v'_a,\dots,v'_{l'}\}$ and let $d\ge a$ be minimal such that
  $v'_d=v_b$. Then $(v_{a-1},\dots,v_b,v'_{d-1},\dots,v'_a)$ is a cycle
  in $I$
  of length $2s\le2k$.

  $(f)\Rightarrow(a)$ holds by Theorem~\ref{sigma:ldc}$(b)$: If each
  $(\alpha,\beta)\in\WR_{k}^I$ satisfies $\alpha=\beta$, then
  Eq.~\eqref{gon:expand2} shows that 
  $\Psi_I(\epsilon,z)$ as defined in
  Eq.~\eqref{gon:psi} is constant in $\epsilon$.
\end{proof}

\begin{rem}
  The equivalence $(b)\Leftrightarrow(g)$ is a noncommutative
  analogue to \cite[Prop.~2.5$(ii)$]{ne98}.
\end{rem}

\begin{rem}
  In \cite[Th.~2.7]{ne99}, the condition of Th.~\ref{sigma:THM}$(f)$
  is visualised in another way: a closed walk
  $P=(c_1,r_1,\dots,c_s,r_s)\sim(\alpha,\beta)$ in $\N\times\N$ is
  considered as the polygonal closed curve $\gamma$ in $\C$ with sides
  parallel to the coordinate axes whose successive vertices are
  $r_1+\iu c_1$, $r_1+\iu c_2$, $r_2+\iu c_2$, \dots, $r_{s-1}+\iu
  c_s$, $r_s+\iu c_s$, $r_s+\iu c_1$ and again $r_1+\iu c_1$. Then
  $\alpha=\beta$ if and only if the index with respect to $\gamma$ of
  every point not on $\gamma$ is zero, if and only if $\gamma$ can be
  shrunk to a point inside of the set of its points. 
\end{rem}

\begin{rem} 
  One cannot drop the assumption that $x$ has real matrix
  coefficients in Th.~\ref{sigma:THM}$(e)$. Consider a $2\times2$
  matrix $x$. Then $\tr x^*x=\sum|x_q|^2$ and $\det
  x^*x=|x_{00}x_{11}-x_{01}x_{10}|^2$.  This shows that if
  $\Re(\overline{x_{00}x_{11}}x_{01}x_{10})=0$, e.g.\ $x=
  \begin{pmatrix}
    1&1\\1&\iu
  \end{pmatrix}
  $, then the singular values of $x$ do not depend on the real sign of
  the matrix coefficients of $x$, whereas
  $(\text{col}\,0,\text{row}\,0,\text{col}\,1,\text{row}\,1)$ is a
  cycle of length 4.
\end{rem}

\begin{rem}
  Theorem~\ref{sigma:THM}$(h)\Rightarrow(a)$ is the isometric counterpart to
  \cite[Th.\ 3.1]{hno01}, which shows in particular that $I$ is an
  unconditional basic sequence in $\Sch^{2k}$ if the number of
  walks in $I$ between two given vertices of length $k$ and with no edge repeated
  has a uniform bound. The following combinatorial problem arises
  naturally: if $I$ satisfies this latter condition, is it
  so that $I$ is the union of a finite number of sets $I_j$ such that there
  is at most one path of length at most $k$ in $I_j$ between two given
  vertices? In the simplest case, $k=2$, William Banks, Ilijas Farah,
  Asma Harcharras and Dominique
  Lecomte \cite{fl03} have deduced from \cite{ro85} that it is not so.
\end{rem}

\section{\texorpdfstring{Metric unconditional approximation pro\-per\-ty for $\Sch^p_I$}{Metric unconditional approximation property for entry subspaces of a Schatten class}}\label{ss10}

\noindent Let $\Row,\Col$ be two copies of $\N$. It is well known that, apart
from $\Sch^2$, no $\Sch^p$
has an unconditional basis or just a local unconditional structure
(see \cite[\S\,4]{px03}.) $\Sch^1$ and $\Sch^\infty$ cannot even be
embedded in a space with unconditional basis. If $1<p<\infty$, then
$\Sch^p$ has the unconditional finite dimensional decomposition
\begin{equation*}
  \bigoplus_{n\in\N}\Sch^p_{\{(r,c):r\le n,c=n\}}\oplus
  \Sch^p_{\{(r,c):r=n+1,c\le n\}}
\end{equation*}
because the triangular projection associated to the idempotent Schur multiplier
$(\chi_{r\le c})$ is bounded on $\Sch^p$.

\begin{dfn}\label{muap:def}
  Let $X$ be a separable Banach space and $\Sign=\T$
  (vs.~$\Sign=\{-1,1\}$.)

  \begin{itemize}
  \item A sequence $(T_k)$ of operators on $X$ is an
    \emph{approximating sequence} if each $T_k$ has finite rank and
    $\|T_kx-x\|\to0$ for every $x\in X$. 
    An approximating sequence of commuting projections
    is a \emph{finite-dimensional decomposition}.
  \item (\cite{pw71}.) The \emph{difference sequence}
    $(\Delta T_k)$ of $(T_k)$ is given by $\Delta T_1=T_1$ and $\Delta
    T_k=T_k-T_{k-1}$ for $k\ge2$. $X$ has the \emph{unconditional
      approximation property} $(uap)$ if there is an approximating
    sequence $(T_k)$ such that for some constant~$D$
    \begin{equation*}
      \biggl\|\sum_{k=1}^n\epsilon_k\Delta T_k\biggr\|
      \le D\quad\text{for all $n$ and $\epsilon_k\in\Sign$}.
    \end{equation*}
    The \emph{complex \klammer{vs.~real} unconditional constant} of
    $(T_k)$ is the least such constant~$D$.
  \item (\cite[\S\,3]{ck91}, \cite[\S\,8]{gks93}.) $X$ has the
    \emph{complex \klammer{vs.~real} metric unconditional approximation
      property} $(muap)$ if, for every $\delta>0$, $X$ has an approximating
    sequence with complex (vs.~real) unconditional constant
    $1+\delta$. By \cite[Th.\ 3.8]{ck91} and \cite[Lemma 8.1]{gks93},
    this is the case if and only if there is an approximating sequence
    $(T_k)$ such that
    \begin{equation}\label{block:muap:def}
      \sup_{\epsilon\in\Sign}\|T_k+\epsilon (\Id-T_k)\|\longrightarrow1.
    \end{equation}
  \end{itemize}

\end{dfn}

$X$ has $(muap)$ if and
only if, for every given $\delta>0$, $X$ is isometric to a
1-complemented subspace of a space with a $(1+\delta)$-unconditional
finite-dimensional decomposition \cite[Cor.~IV.4]{gk97}. If $X$ has
$(muap)$, then, for any given $\delta>0$, $X$ is isometric to a
subspace of a space with a $(1+\delta)$-unconditional basis.

\begin{exa}
  The simplest example is the subspace in $\Sch^p$ of operators with an
  upper triangular matrix. In fact, if $I\subseteq\Row\times\Col$ is
  such that all columns $I\cap\Row\times\{c\}$ (vs.~all rows
  $I\cap\{r\}\times\Col$) are finite, then $\Sch^p_I$ admits a
  1-un\-con\-di\-tio\-nal finite-dimensional decomposition in the
  corresponding finitely supported idempotent Schur multipliers
  $\chi_{I\cap\Row\times\{c\}}$ (vs.~$\chi_{I\cap\{r\}\times\Col}$.)
\end{exa}

Our results on complete 1-un\-con\-di\-tio\-nal basic sequences yield the
following theorem.

\begin{thm}
  Let $1\le p\le\infty$. Let $\Row_r\subseteq\Row$, $r\in\N$, be pairwise
  disjoint and finite. Let $\Col_c\subseteq\Col$, $c\in\N$, be pairwise
  disjoint and finite. Let $J\subseteq\N\times\N$ and
  $I=\bigcup_{(r,c)\in J}\Row_r\times\Col_c$. Then the sequence of Schur
  multipliers $(\chi_{\Row_r\times\Col_c})_{(r,c)\in J}$ forms a
  complex 1-un\-con\-di\-tio\-nal finite-dimensional
  decomposition for $\Sch^p_I$ if and only if $J$ is a forest or $p$ is
  an even integer and $J$ contains no cycle of length $4,6,\dots,p$.
\end{thm}

We may always suppose that approximating sequences on spaces $\Sch^p_I$
are associated to Schur multipliers. More precisely, we have

\begin{prp}
  Let $1\le p\le\infty$ and $I\subseteq\Row\times\Col$.  Let $(T_n)$ be an
  approximating sequence on $\Sch^p_I$. Then there is a sequence of Schur
  multipliers $(\varphi_n)$ such that $(\Schur_{\varphi_n})$ is
  an approximating sequence on $\Sch^p_I$ and such that if 
  $(T_n)$ satisfies \eqref{block:muap:def}, then so does
  $(\Schur_{\varphi_n})$.
\end{prp}


\begin{proof}
  Let $\delta_n>0$ be such that $\delta_n\to0$.  As $T_n$ has finite
  rank, there is a finite $\Row_n\times\Col_n\subseteq\Row\times\Col$
  such that the projection $P_{\Row_n\times\Col_n}$ of $\Sch^p$
  onto $\Sch^p_{\Row_n\times\Col_n}$ defined by the Schur
  multiplier $\chi_{\Col_n}\otimes\chi_{\Row_n}$ satisfies
  $\|P_{\Row_n\times\Col_n}T_n-T_n\|<\delta_n$. 
  Let $\varphi_n$ be the Schur multiplier
  associated to $[T_n]_{\Row_n\times\Col_n}$. 
  With the notation of Eq.~\eqref{ave},
  \begin{equation*}
    M_{\varphi_n}(x) - x = \int_{\T^{\Row}}\mathrm{d}\eta\,
    \int_{\T^{\Col}}\mathrm{d}\zeta\,
    \Schur_{\zeta^*\otimes\eta^*}
    (P_{\Row_n\times\Col_n}T_n-\Id)(\Schur_{\zeta\otimes\eta}x).
  \end{equation*}
  As $P_{\Row_n\times\Col_n}T_n$ tends to the identity uniformly on
  compact sets, this shows that $M_{\varphi_n}$ is an approximating
  sequence.  As \[ \Schur_{\varphi_n} + \epsilon
  (\Id-\Schur_{\varphi_n}) = [P_{\Row_n\times\Col_n}T_n + \epsilon
  (\Id-P_{\Row_n\times\Col_n}T_n)]\ ,\] the norm of this operator is
  at most $\|T_n+\epsilon(\Id-T_n)\|+2\delta_n$.
\end{proof}

This proposition shows together with Prop.~\ref{pinf} the following results.

\begin{cor}
  Let $1\le p\le\infty$ and $I\subseteq\Row\times\Col$. 

  \begin{itemize}
  \item If $\Sch^p_I$ has $(muap)$, then  some
    sequence of Schur multipliers realises it.
  \item Let $J\subseteq I$. If $\Sch^p_I$ has $(muap)$, then so does $\Sch^p_J$.
  \item If $\Sch^\infty_I$ has $(muap)$, then so does $\Sch^p_I$.
  \end{itemize}

\end{cor}

Let us define the following asymptotic properties.

\begin{dfn}
    Let $1\le p\le\infty$, $I\subseteq\Row\times\Col$ and $\Sign=\T$
  (vs.~$\Sign=\{-1,1\}$.) 
  \begin{itemize}
  \item $\Sch^p_I$ is \emph{asymptotically unconditional} 
    if for every $x\in\Sch^p_I$ and for every bounded sequence
    $(y_n)$ in $\Sch^p_I$ such that each matrix coefficient of
    $y_n$ tends to 0
    \begin{equation*}
      \max_{\epsilon\in\Sign}\|x+\epsilon y_n\|_p -
      \min_{\epsilon\in\Sign}\|x+\epsilon y_n\|_p\longrightarrow0.
    \end{equation*}
  \item $I$ enjoys the property $(\mathscr{U})$ of \emph{block
    unconditionality} in $\Sch^p$ if for each $\delta>0$ and finite $F\subseteq I$, there
    is a finite $G\subseteq I$ such that 
    \begin{equation*}\forall\,x\in B_{\Sch^p_F}~\forall\,y\in
    B_{\Sch^p_{I\setminus G}}\quad 
      \max_{\epsilon\in\Sign}\|x+\epsilon y\|_p -
      \min_{\epsilon\in\Sign}\|x+\epsilon y\|_p < \delta.
    \end{equation*}
  \end{itemize}
\end{dfn}

The arguments of \cite[\S\,6.2]{ne98} show mutatis mutandis

\begin{thm}\label{muap}
  Let $1\le p\le\infty$, $I\subseteq\Row\times\Col$ and $\Sign=\T$
  \klammer{vs.~$\Sign=\{-1,1\}$.} Consider the following properties.
  
  \begin{enumerate}
  \item $\Sch^p_I$ is asymptotically unconditional.
  \item $I$ enjoys $(\mathscr{U})$ in $\Sch^p$.
  \item $\Sch^p_I$ has $(muap)$.
  \end{enumerate}

  Then $(c)\Rightarrow(a)\Leftrightarrow(b)$. If $1<p<\infty$, then $(b)\Leftrightarrow(c)$. If
  $p=1$, $\Sch^1_I$ has $(muap)$ if and only if $\Sch^1_I$ has $(uap)$ and
  $I$ enjoys $(\mathscr{U})$ in $\Sch^1$.
\end{thm}

The case $p=\infty$ is extreme in the sense that the following properties are
equivalent for $\Sch^\infty_I$: to be a dual space, to be reflexive, to
have a finite
cotype, not to contain $\cnaught$, because they are equivalent for $I$ not to contain
any sequence $(r_n,c_n)$ with $(r_n)$ and $(c_n)$ injective, that
is for $I$ to be
contained in the union of a finite set of lines and a finite set of columns,
so that $\Sch^\infty_I$ is isomorphic to $\ell^2_I$.

Let us now introduce the asymptotic property on $I$ that reflects the
combinatorics imposed by $(muap)$.

\begin{dfn}
  Let $I\subseteq\Row\times\Col$ and $k\ge1$. 

  \begin{itemize}
  \item $I$ enjoys property $\mathscr{J}_k$ if for every path
    $P=(c_0,r_0,\dots,c_j,r_j)$ of odd length $2j+1\le k$ in $I$ there
    is a finite set $\Row'\times\Col'$ such that $P$ cannot be completed
    with edges in $I\setminus\Row'\times\Col'$ to a cycle of length
    $2s\in\{4j+2,\dots,2k\}$.
  \item The \emph{asymptotic distance} $d_\infty(r,c)$ of
    $r\in\Row$ and $c\in\Col$ in $I$ is the supremum, over all finite
    rectangle sets $\Row'\times\Col'$, of the distance from $r$ to $c$
    in $I\setminus\Row'\times\Col'$.
  \end{itemize}

\end{dfn}

The asymptotic distance takes its values in $\{1,3,5,\dots,\infty\}$.
Note that $\mathscr{J}_1$ is true and that
$\mathscr{J}_k\Rightarrow\mathscr{J}_{k-1}$. This implication is strict: let $\Row,\Col$
be two copies of $\N$ and, given $j\ge1$, consider the union $I_j$ of
all paths $(\text{col}\,0,\allowbreak \text{row}\,{nj+1},\allowbreak
\text{col}\,{nj+1},\allowbreak \dots,\allowbreak
\text{row}\,{nj+j},\allowbreak \text{col}\,{nj+j},\allowbreak
\text{row}\,0)$ of length $2j+1$. Then $I_j$ contains no cycle of
length $2s\in\{4,\dots,4j\}$ and therefore enjoys $\mathscr{J}_{2j}$,
but fails $\mathscr{J}_{2j+1}$;
$I_j\cup\{(\text{row}\,0,\text{col}\,0)\}$ contains no cycle of length
$2s\in\{4,\dots,2j\}$ and thus enjoys $\mathscr{J}_{j}$, but
fails $\mathscr{J}_{j+1}$. In particular, the properties
$\mathscr{J}_k$, $k\ge2$, are not stable under union with a
singleton. 

Let us now explicit the relationship between $\mathscr{J}_k$ and $d_\infty$.

\begin{prp}
  Let $I\subseteq\Row\times\Col$ and $k\ge1$. 

  \begin{enumerate}
  \item $I$ enjoys $\mathscr{J}_k$ if and only if any two
    vertices $r\in\Row$ and $c\in\Col$ at distance $2j+1\le k$
    satisfy $d_\infty(r,c)\ge2k-2j+1$.
  \item If $d_\infty(r,c)\ge2k+1$ for all $(r,c)\in\Row\times\Col$,
    then $I$ enjoys $\mathscr{J}_k$.
  \item If $d_\infty(r,c)\le k$ for some $(r,c)\in\Row\times\Col$,
    then $I$ fails $\mathscr{J}_k$.
  \item $I$ enjoys $\mathscr{J}_k$ for every $k$ if and only if
    $d_\infty(r,c)=\infty$ for every $(r,c)\in\Row\times\Col$.
  \end{enumerate}

\end{prp}

\begin{proof}
  $(a)$ is but a reformulation of the definition of$\mathscr{J}_k$ and
  implies $(b)$. 

  $(d)$ is a consequence of $(b)$ and $(c)$.
  
  $(c)$. If $d_\infty(r,c)\le k$, then there is $0\le j\le(k-1)/2$
  such that there are infinitely many paths of length $2j+1$ from $c$
  to $r$: there is a path $(c,\allowbreak r_1,\allowbreak
  c_1,\dots,\allowbreak r_j,\allowbreak c_j,\allowbreak r)$ that can
  be completed with edges outside any given finite set to a cycle of
  length $4j+2\le2k$.
\end{proof}

\begin{thm}
  Let $I\subseteq\Row\times\Col$ and $1\le p\le\infty$. If $p$ is an even
  integer, then $\Sch^p_I$ has complex or real $(muap)$ if and only if
  $I$ enjoys $\mathscr{J}_{p/2}$. If $p=\infty$ or if $p$ is not an even
  integer, then $\Sch^p_I$ has real $(muap)$ only if
  $I$ enjoys $\mathscr{J}_k$ for every $k$.
\end{thm}

\begin{proof}
  Suppose that $I$ enjoys $(\mathscr{U})$ in $\Sch^p$ and fails $\mathscr{J}_k$.
  Then, for some $s\le k$, $I$ contains a sequence of cycles $(c_0,\allowbreak
  r_0,\dots,\allowbreak c_{j-1},\allowbreak r_{j-1},\allowbreak
  c_j^n,\allowbreak r_j^n,\dots,\allowbreak c_{s-1}^n,\allowbreak
  r_{s-1}^n)$
  with the property that $\|x-y\|_p\le(1+1/n)\|x+y\|_p$ for all $x$ with support
  in $\{(r_0,c_0),\allowbreak(r_0,c_1),\dots,\allowbreak
  (r_{j-2}, c_{j-1}),\allowbreak (r_{j-1}, c_{j-1})\}$ and all $y$ with
  support in $\{(r_{j-1}, c_j^n),\allowbreak
  (r_j^n, c_j^n),\allowbreak \dots,\allowbreak (r_{s-1}^n, c_{s-1}^n),\allowbreak
  (r_{s-1}^n, c_0)\}$. With the notation of Section~\ref{sec:cst}, this
  amounts to stating that the multiplier on
  $I=\{(i,i),(i,i+1)\}\subseteq\Z/s\Z\times\Z/s\Z$ given by
  $\epsilon_{rc}=1$ if $r,c\in\{0,\dots,j-1\}$ and $\epsilon_{rc}=-1$
  otherwise actually is an isometry on $\Sch^p_I$. As
  $\overline{\epsilon_{00}} \epsilon_{01} \dots
  \overline{\epsilon_{{s-1}\,{s-1}}}
  \epsilon_{{s-1}\,0}=(-1)^{2s-2j+1}=-1$, this implies by
  Prop.~\ref{cyno}$(e)$ that $p/2\in\{1,2,\dots,s-1\}$.
  
  Suppose that $I$ enjoys $\mathscr{J}_k$. We claim that for every finite
  $F\subseteq I$ there is a finite $G\subseteq I$ such that every
  closed walk $P\sim(\alpha,\beta)$ of length $2k$ in $I$ satisfies
  $\sum_{q\in I\setminus G}\beta_q-\alpha_q=0$. This signifies that given a
  closed walk $(v_0,\dots,v_{2k-1})$ and
  $0=a_0<b_0<\dots<a_m<b_m<a_{m+1}=2k$ such that
  $v_{a_i},\dots,v_{b_i-1}\in I\setminus G$ and
  $v_{b_i},\dots,v_{a_{i+1}-1}\in F$,
  \begin{equation*}
    \bigl\{i\in\{0,\dots,m\}:a_i,b_i~\text{even}\bigr\} =
    \bigl\{i\in\{0,\dots,m\}:a_i,b_i~\text{odd}\bigr\}.
  \end{equation*}
  Suppose that this
  is not true: then there is an $s\le k$, there are
  $0=a_0<b_0<\dots<a_m<b_m<2s$ and there are cycles
  $(v_{a_0}^n,\dots,\allowbreak v_{b_0-1}^n,\allowbreak
  v_{b_0},\dots,\allowbreak v_{a_1-1},\dots,\allowbreak
  v_{a_m}^n,\dots,\allowbreak v_{b_m-1}^n,\allowbreak
  v_{b_m},\dots,\allowbreak v_{2s-1})$ such that the $(v_i^n)_{n\ge0}$
  are injective sequences of vertices and $b_i-a_i$ is even for at
  least one index $i$: let us suppose so for $i=0$. If $b_0-a_0\ge
  s-1$, consider the path
  $P=(v_{b_0},\dots,\allowbreak v_{a_0-1},\allowbreak
  v_{a_0}^0,\dots,\allowbreak v_{b_m-1}^0,\allowbreak
  v_{b_m},\dots,\allowbreak v_{2s-1})$
  of odd length $2s-1-(b_0-a_0)$; if $b_0-a_0\le s-1$, consider
  the path $P=(v_{2s-1},\allowbreak v_{a_0}^0,\dots,\allowbreak
  v_{b_0-1}^0,\allowbreak v_{b_0})$ of odd
  length $b_0-a_0+1$. Then $P$ can be completed with vertices
  outside any given finite set to a cycle of length at most $2s$
  because $(v_{2s-1},v_{a_0}^n,\dots,\allowbreak
  v_{b_0-1}^n,\allowbreak v_{b_0})$ is a path of length $b_0-a_0+1$ in
  $I$ for every $n$. This proves that $I$ fails $\mathscr{J}_s$. 
  
  The claim shows that $I$ enjoys $(\mathscr{U})$ in $\Sch^p$ for $p=2k$. In
  fact, if $\tilde\epsilon\in\T^{F\cup(I\setminus G)}$ is defined by
  $\tilde\epsilon_q=1$ for $q\in F$ and
  $\tilde\epsilon_q=\epsilon\in\T$ for $q\in I\setminus G$, then, with
  the notation of Th.~\ref{sigma:ldc},
  \begin{equation*}
    \Phi_{F\cup(I\setminus G)}(\tilde\epsilon,a) =
    \sum_{(\alpha,\beta)\in\WR_{k}^{F\cup(I\setminus G)}}
    n_{\alpha\beta}\epsilon^{\sum_{q\in I\setminus G}
      \beta_q-\alpha_q}\overline{a}^\alpha a^\beta
  \end{equation*}
  does not depend on
  $\epsilon$, so that $\|x+\epsilon y\|_{2k}=\|x+y\|_{2k}$ if $x\in
  \Sch^{2k}_F$ and $y\in \Sch^{2k}_{I\setminus G}$, and $\Sch^{2k}_I$ has
  complex $(muap)$ by Th.~\ref{muap}$(b)\Rightarrow(c)$.
\end{proof}

\begin{rem}
  This theorem is a noncommutative analogue to \cite[Th.~7.5]{ne98}.
\end{rem}

\section{Examples}\label{ss11}

\noindent One of Varopoulos' motivations for the study of the projective tensor
product $\ell_\infty\projtens\ell_\infty$ are lacunary sets in a
locally compact abelian group. 

Let $\Gamma$ be a discrete abelian group and $\Lambda\subseteq\Gamma$.
Let us say that $\Lambda$ is \emph{$n$-independent} if every element
of $\Gamma$ admits at most one representation as the sum of $n$
terms in $\Lambda$, up to a permutation. For example, the geometric
sequence $\{j^k\}_{k\ge0}$ with $j\in\{2,3,\dots\}$ is $n$-independent
in $\Z$ if and only if $j\ge n$ \cite[\S\,3]{ne98}. If $\Lambda$ is
$n$-independent for all $n$, then $\Lambda$ is \emph{independent}.
Let
\[\Zeta_n=\{\zeta\in\Z^\Lambda:\sum_{\gamma\in\Lambda}\zeta_\gamma=0\
\text{and}\ \sum_{\gamma\in\Lambda}|\zeta_\gamma|\le2n]\] and
$\Zeta=\bigcup\Zeta_n$. Then $\Lambda$ is $n$-independent if and only
if, for every $\zeta\in\Zeta_n$,
\[\sum_{\gamma\in\Lambda}\zeta_\gamma\gamma=0\ \Rightarrow\ \zeta=0\]
and $\Lambda$ is independent if and only if this holds for every
$\zeta\in\Zeta$. 

Let us say that $\Lambda$ is \emph{$n$-independent modulo 2} if in
every representation of an element of $\Gamma$ as the sum of $n$
terms in $\Lambda$, each element of $\Lambda$ appears the same
number of times modulo 2. In other words, for every
$\zeta\in\Zeta_n$,
\[\sum_{\gamma\in\Lambda}\zeta_\gamma\gamma=0\quad
\Longrightarrow\quad \forall\,\gamma\in\Lambda\quad
\zeta_\gamma=0\pmod2\ ;\] $\Lambda$ is \emph{independent modulo
  2} if this holds for every $\zeta\in\Zeta$.
If $\Gamma$ contains no element of order 2, then one may always
suppose that at least one coefficient $\zeta_\gamma$ of a nontrivial
relation $\sum\zeta_\gamma\gamma=0$ is odd, so that these two latter
notions ``modulo 2'' coincide
with the two former ones.

Let $G=\hat\Gamma$, so that $\Gamma$ is the group of characters on
$G$. Then the computation presented in \cite[Prop.~2.5$(ii)$]{ne98}
for the case $\Gamma=\Z$ shows that $\Lambda$ is a complex
(vs.~real) 1-un\-con\-di\-tio\-nal basic sequence in
$\Ell^p(G)$ with $p\in2\N^*$
if and only if $\Lambda$ is $p/2$-independent (vs.~modulo 2).
Furthermore $\Lambda$ is a complex (vs.~real)
1-un\-con\-di\-tio\-nal basic sequence in $\Ell^{p}(G)$ with
$p\in(0,\infty]\setminus2\N^*$ if and only if $\Lambda$ is independent
(vs.~modulo 2). If $\Gamma$ contains no element of order
2, then a real 1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$ is
also complex 1-un\-con\-di\-tio\-nal.  All these results hold also for the
complete counterparts to 1-un\-con\-di\-tio\-nal basic sequences.

Results on lacunary sets in a discrete abelian group transfer to lacunary
matrices in the following way, as in \cite[Th.~4.2]{va68}.

\begin{prp}\label{Gamma}
  Let $\Gamma$ be a discrete abelian group and $\Row$, $\Col$ be
  countable subsets of $\Gamma$. To every
  $\Lambda\subseteq\Row+\Col$ associate
  $I_\Lambda=\{(r,c)\in\Row\times\Col:r+c\in\Lambda\}$. Let
  $G=\hat\Gamma$. 

  \begin{enumerate}
  \item If $\Lambda$ is a complex 1-un\-con\-di\-tio\-nal basic sequence in
    $\Ell^4(G)$, then $I_\Lambda$ is a 1-un\-con\-di\-tio\-nal basic
    sequence in $\Sch^4$.
  \item Suppose that each element of $\Gamma$ admits at most one
    representation as the sum of an element of $\Row$ with an element
    of $\Col$. Then every $I\subseteq\Row\times\Col$ has the form
    $I=I_\Lambda$ with $\Lambda={\{r+c:(r,c)\in I\}}$. If $\Lambda$ is
    a real 1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$, then $I_\Lambda$
    is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item Let $p=2k$ be a positive even integer. Suppose that 
    $\Row\cap\Col=\emptyset$ and $\Row\cup\Col$ is $k$-independent modulo 2. 
    $I_\Lambda$ is a 1-un\-con\-di\-tio\-nal basic
    sequence in $\Sch^p$ if and only if $\Lambda$ is a real
    1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$.
  \end{enumerate}
  
\end{prp}

\begin{proof}
  $(a)$. Let $P=(c,r,c',r')$ be a closed walk in $I_\Lambda$. Then
  $r+c$, $r'+c'$, $r+c'$ and $r'+c$ are in $\Lambda$ while
  $(r+c)+(r'+c')=(r+c')+(r'+c)$: if $\Lambda$ is
  2-independent, then $r+c\in\{r+c',r'+c\}$, so that $c=c'$ or
  $r=r'$ and $P$ is not a cycle. 
  
  $(b)$. For each $\gamma\in\Lambda$, let
  $q_\gamma=(r_\gamma,c_\gamma)$ be the unique element of $I$ such
  that $r_\gamma+c_\gamma=\gamma$. If $\Lambda$ is a real
  1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$, then it is also a
  complete real 1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$. Let
  $\varphi\in\{-1,1\}^{I_\Lambda}$, so that
  $\varphi_{q_\gamma}\in\{-1,1\}$ for all $\gamma\in\Lambda$. Then, as
  in Eq.~\eqref{tr1},
  \begin{multline*}
    \Bigl\| \sum_{q\in I_\Lambda} a_q\e_q
    \Bigr\|_{\Sch^p_{I_\Lambda}(\Sch^p)} = \Bigl\|
    \sum_{(r,c)\in I_\Lambda} r(g)c(g)a_{rc}\e_{rc}
    \Bigr\|_{\Sch^p_{I_\Lambda}(\Sch^p)}\\
    = \biggl\| \sum_{\gamma\in\Lambda}
    a_{q_\gamma}\e_{q_\gamma}\gamma(g)
    \biggr\|_{\Sch^p_{I_\Lambda}(\Sch^p)}
    = \biggl\| \sum_{\gamma\in\Lambda}
    a_{q_\gamma}\e_{q_\gamma}\gamma
    \biggr\|_{\Ell^p_\Lambda(G,\Sch^p(\Sch^p))}
  \end{multline*}
  so that as in Eq.~\eqref{tr2}, by complete real 1-un\-con\-di\-tio\-nality
  of $\Lambda$ in $\Ell^p(G)$,
  \begin{align*}
    \Bigl\| \sum_{q\in I_\Lambda} \varphi_qa_q\e_q
    \Bigr\|_{\Sch^p_{I_\Lambda}(\Sch^p)} = \biggl\|
    \sum_{\gamma\in\Lambda}
    \varphi_{q_\gamma}a_{q_\gamma}\e_{q_\gamma}\gamma
    \biggr\|_{\Ell^p_\Lambda(G,\Sch^p(\Sch^p))}= \Bigl\|
    \sum_{q\in I_\Lambda} a_q\e_q
    \Bigr\|_{\Sch^p_{I_\Lambda}(\Sch^p)}.
  \end{align*}

  $(c)$. Each element of $\Gamma$ admits at most one
    representation as the sum of an element of $\Row$ with an element
    of $\Col$, so that $(b)$ yields sufficiency. Suppose that $\Lambda$ is not a real
  1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$ and let
  $\zeta\in\Zeta_k$ such that $\sum_{\gamma\in\Lambda}\zeta_\gamma\gamma=0$ and
  $J=\{(r,c)\in I_\Lambda:\zeta_{r+c}\ne0\pmod2\}$ is nonempty; $J$
  has at most $2k$ elements. 
  Let $P=(v_1,\dots,v_j)$ be a path in $J$ of maximal length. Then
  $\zeta_{v_{j-1}+v_j}$ is odd and
  $\sum\{\zeta_{v_j+v}:v_j+v\in\Lambda\}$ is even because it is the
  coefficient of $v_j$ in the relation $\sum_{\gamma\in\Lambda}\zeta_\gamma\gamma=0$ and
  $\Row\cup\Col$ is $k$-independent modulo 2. There is therefore $v_{j+1}$ distinct from
  $v_{j-1}$ such that $\zeta_{v_j+v_{j+1}}$ is odd. As $j$ is maximal
  and $\Row\cap\Col=\emptyset$, $v_{j+1}=v_{j+1-2i}$ for
  some $2\le i\le k$, so that $(v_{j+1-2i},\dots,v_j)$ is a cycle of
  length $2i$ in $J$: $I_\Lambda$ is not a 1-un\-con\-di\-tio\-nal basic
    sequence in $\Sch^p$. 
\end{proof}

Let $\Row$ and $\Col$ be any countable sets. Consider
$G=\{-1,1\}^\Col\times\{-1,1\}^\Row$. If we denote by
$\bigl((\epsilon_c)_{c\in\Col},\allowbreak(\epsilon'_r)_{r\in\Row}\bigr)$ a
generic point in $G$, then the set of Rademacher functions
$\{\epsilon_c\}_{c\in\Col}\cup\{\epsilon'_r\}_{r\in\Row}$ is a real
1-unconditional basic sequence in $\Cont(G)$, so that it is
independent modulo 2 in $\hat G$.
Similarly, the set of Steinhaus functions
$\{z_c\}_{c\in\Col}\cup\{z'_r\}_{r\in\Row}$ is independent in the dual
of $\T^\Col\times\T^\Row$.
This yields:

\begin{cor}\label{cor:S}
  Let $I\subseteq\Row\times\Col$ and $p\in(0,\infty]$. The following
  are equivalent:

  \begin{itemize}
  \item $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item $\{\epsilon_c\epsilon'_r:(r,c)\in I\}$ is a real
    1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$.
  \item $\{z_cz'_r:(r,c)\in I\}$ is a 
    1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(\T^\Col\times\T^\Row)$.
\end{itemize}

\end{cor}

\begin{rem}
  The isomorphic counterpart is also true: $I$ is a completely
  unconditional basic sequence in $\Sch^p$ (i.e., a
  complete $\sigma(p)$ set) if and only if
  $\{\epsilon_c\epsilon'_r:(r,c)\in I\}$ is a completely
  unconditio\-nal basic sequence in $\Ell^p(G)$ (a
  \emph{$\Lambda(p)_{\mathrm{cb}}$ set} in $\hat G$, see \cite{ha98}
  and \cite[\S\,8.1]{pi98},) if and only if $\{z_cz'_r:(r,c)\in I\}$
  is a completely unconditio\-nal basic sequence in
  $\Ell^p(\T^\Col\times\T^\Row)$.  This follows e.g.\ from the
  proof of Prop.~\ref{Gamma}$(b)$ and the iterated noncommutative
  Khinchin inequality \cite[Eq.\ (8.4.11)]{pi98}.
\end{rem}

Harcharras \cite{ha98} used Peller's discovery \cite{pe82a} of
the link between Fourier and Hankel Schur multipliers to produce
unconditional basic sequences in $\Sch^p$ that are unions of
antidiagonals in $\N\times\N$.  We have in our context the rather disappointing

\begin{prp}\label{hoory}
  Let $\Lambda\subseteq\N\subseteq\Z$ and
  $I=\{(r,c)\in\N\times\N:r+c\in\Lambda\}$.
  
  \begin{enumerate}
  \item $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^4$ if
    and only if $\{z^\lambda\}_{\lambda\in\Lambda}$ is a
    1-un\-con\-di\-tio\-nal basic sequence in $\Ell^4(\T)$. 
  \item If $\Lambda$ contains three elements $\lambda<\mu<\nu$ such that
    $\lambda+\mu\ge\nu$, then $I$ is not a
    1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$ if
    $p\in(0,\infty]\setminus\{2,4\}$.
    \item If $\Lambda=\{\lambda_k\}$ with $\lambda_{k+1}>2\lambda_k$
    for all $k$,
    then $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$
    for every $p$. 
  \end{enumerate}
  
\end{prp}

\begin{proof} $(a)$. Sufficiency follows from Prop.~\ref{Gamma}$(a)$ with
    $\Row=\Col=\N$. Conversely, if $\Lambda$ contains a solution to
  $\lambda+\mu=\lambda'+\mu'$ with $\lambda<\lambda'\le \mu'<\mu$, then $I$ contains the cycle
  $(\text{col}\,0,\text{row}\,\lambda,\text{col}\,\lambda'-\lambda,\text{row}\,\mu')$. 

$(b)$. Consider the cycle $(\text{col}\,0, \text{row}\,\lambda,
\text{col}\,\nu-\lambda,
\text{row}\,\mu-\nu+\lambda,
\text{col}\,\nu-\mu, \text{row}\,\mu)$. 

$(c)$. In fact, $I$ is a forest. Let $P=(c_1,r_1,\dots,c_k,r_k)$ be a
closed walk in $I$. We may suppose without loss of generality that
$r_1+c_2$ is a maximal element of
$\{r_1+c_1,r_1+c_2,\dots,r_k+c_k,r_k+c_1\}$. Then $r_1+c_1\le r_1+c_2$
and $r_2+c_2\le r_1+c_2$. One of these inequalities must be an
equality and $P$ is not a cycle: for otherwise $2(r_1+c_1)<r_1+c_2$
and $2(r_2+c_2)<r_1+c_2$ because $r_1+c_1,r_1+c_2,r_2+c_2\in\Lambda$,
so that $2(r_1+c_1+r_2+c_2)<2(r_1+c_2)$ and $c_1+r_2<0$.
\end{proof}

\begin{rem}
  Further computations yield the following result. If
  $\{z^\lambda\}_{\lambda\in\Lambda}$ is a 1-un\-con\-di\-tio\-nal
  basic sequence in $\Ell^6(\T)$ and if
  $\{\lambda<\mu<\nu\}\subseteq\Lambda\Rightarrow\lambda+\mu<\nu$,
  then $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in
  $\Sch^6$; the converse does not hold.
\end{rem}

Let us now give an overview of the known extremal bipartite graphs
without cycle of length $4,6,\dots,2k$ and their size.  Look up
\cite[Def.~I.3.1]{bjl99} for the definition of a Steiner system and
\cite[Def.~1.3.1]{va98} for the definition of a generalised
polygon. An  elementary example is given in the introduction with \eqref{steiner}. 

\begin{prp}\label{extr:prp}
  Let $2\le n\le m$, $I\subseteq\Row\times\Col$ with $\card\Col=n$ and $\card\Row=m$,
  and $e=\card I$.

  \begin{enumerate}
  \item If $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^4$,
    then 
    \begin{equation*}n\ge1 + \Bigl(\frac em-1\Bigr) + \Bigl(\frac
      em-1\Bigr)\Bigl(\frac en-1\Bigr),\end{equation*}
    that is
    $e^2-me-mn(n-1)\le0$. Equality holds if and only if $I$ is the
    incidence graph of a Steiner system $\mathrm S(2,e/m;n)$ on $n$ points and
    $m$ blocks.
  \item If $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^6$,
    then 
    \begin{equation*}n\ge1 + \Bigl(\frac em-1\Bigr) + \Bigl(\frac
      em-1\Bigr)\Bigl(\frac en-1\Bigr) + \Bigl(\frac
      em-1\Bigr)^2\Bigl(\frac en-1\Bigr),\end{equation*}
    that is
    $e^3-(m+n)e^2+2mne-m^2n^2\le0$. Equality holds if and only if $I$ is
    the incidence graph of the \emph{quadrangle} \klammer{the cycle of length 8} or
    of a generalised quadrangle with $n$ points and $m$ lines.
  \item If $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^{2k}$
    with $k\ge1$ an integer, then 
    \begin{equation}\label{hoo}
      n\ge\sum_{i=0}^k\Bigl(\frac em-1\Bigr)^{\lceil\frac i2\rceil}\Bigl(\frac
      en-1\Bigr)^{\lfloor\frac i2\rfloor}.
    \end{equation}
    Equality holds if $I$ is the incidence graph of the \emph{$(k+1)$-gon} \klammer{the
      cycle of length $2k+2$} or of a generalised $(k+1)$-gon with $n$
    points and $m$ lines.
  \end{enumerate}

\end{prp}

\begin{proof}
  By Theorem~\ref{sigma:THM}$(b)\Rightarrow(g)$, $I$ is a 1-un\-con\-di\-tio\-nal basic
  sequence in $\Sch^{2k}$, with $k\ge1$ an integer, if and only if $I$ is a
  graph of girth $2k+2$ in the sense of \cite{ho02}. Therefore $(a)$ and $(b)$ are shown in
  \cite[Prop.~4, Th.~8, Rem.~10]{ne01}. Inequality \eqref{hoo} is
  \cite[Eq.~(1)]{ho02} and the sufficient condition for equality follows from 
  \cite[Lemma 1.5.4]{va98}. 
\end{proof}

Consult \cite[Tables~A1.1, A5.1]{bjl99} for examples of Steiner
systems and \cite[Table~2.1]{va98} for examples of generalised
polygons. In both cases, the corresponding incidence graph is biregular: every
vertex in $\Row$ has same degree $s+1$ and every vertex in $\Col$ has same
degree $t+1$. Arbitrarily large generalised $(k+1)$-gons exist only
if $2k\in\{4,6,10,14\}$ \cite[Lemma 1.7.1]{va98}; for
$2k\in\{6,10,14\}$, it follows from \cite[Lemma 1.5.4]{va98} that 
\begin{equation*}
  n=(s+1)\frac{(st)^{(k+1)/2}-1}{st-1},~
  m=(t+1)\frac{(st)^{(k+1)/2}-1}{st-1}.
\end{equation*} 

\begin{rem}\label{extr:rem}
  Let $I\subseteq\Row\times\Col$ with $\card\Col=\card\Row=n$. Inequality~\eqref{hoo} shows
  that if $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^{2k}$, then
  $\card I\le n^{1+1/k}+(s-1)n/s$. If $p\notin\{4,6,10\}$, the existence
  of 1-un\-con\-di\-tio\-nal basic sequences in $\Sch^{2k}$ such that
  $\card I\succcurlyeq n^{1+1/k}$ is in fact an important open problem in graph
  theory: extremal graphs cannot correspond to generalised polygons
  and necessarily have less structure.
\end{rem}

\begin{acknowledgements}
  I would like to thank \'Eric Ricard for his comments and suggestions.
\end{acknowledgements}





\noindent
\emph{Keywords.} Schatten-von-Neumann class, Schur product, graph with a
given girth, 1-unconditional basic sequence, metric unconditional
approximation property, V-Sidon set, lacunary set. 

\bigskip

\noindent 
2000 \emph{Mathematics Subject Classification.} 47B10, 46B15, 46B04,
43A46, 05C38, 46B28.

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "h"
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\clearpage

\chapter[Matrix inequalities with applications
to iterated kernels]{Matrix inequalities with applications
to the theory of iterated kernels}\label{miwatik}
\begin{flushright}
  \large with William Banks, Asma Harcharras and \'Eric Ricard. 
\end{flushright}\smallskip

% \noindent{\sc William Banks} \\
% {Department of Mathematics, University of Missouri} \\
% {Columbia, MO 65211 USA} \\
% {bbanks@math.missouri.edu} 

% \noindent{\sc Asma Harcharras} \\
% {Department of Mathematics, University of Missouri} \\
% {Columbia, MO 65211 USA} \\
% {harchars@math.missouri.edu} 

% \noindent{\sc Stefan Neuwirth} \\
% {Laboratoire de Math\'ematiques, Universit\'e de Franche-Comt\'e} \\
% {F-25030 Besan\c con Cedex, France} \\
% {neuwirth@math.univ-fcomte.fr}

% \noindent{\sc \'Eric Ricard} \\
% {Universit\'e Paris VI, Equipe d'Analyse, Case 186} \\
% {75252 Paris Cedex 05, France} \\
% {ericard@ccr.jussieu.fr}


\begin{abstract}
\noindent For an $m\times n$ matrix $A$ with nonnegative real entries,
Atkinson, Moran and Watterson proved the inequality $s(A)^3 \le mn s(AA^tA)$,
where $A^t$ is the transpose of $A$, and $s(\cdot)$ is the sum of the
entries.  We extend this result to finite products of the form
$AA^tAA^t\ldots A$ or $AA^tAA^t\ldots A^t$ and give some applications 
to the theory of iterated
kernels.
\end{abstract}



%%%%%%%%%%%%%%%%
% PAPER BEGINS %
%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%
% INTRODUCTION %
%%%%%%%%%%%%%%%%

\section{Introduction}
\label{sec:intro}

\noindent For any matrix $A$, let $s(A)$ denote the sum of its
entries.  For any integer $k\geq 1$, we define
$$A^{(2k)}=(AA^t)^k,\qquad A^{(2k+1)}=(AA^t)^k A,$$
where $A^t$ denotes the transpose of $A$.  In
Section~\ref{sec:matrix}, we prove the following sharp
inequalities:

\begin{thm}\label{thm1}
  Let $A$ be an $m\times n$ matrix with nonnegative real entries.
  Then for every integer $k\ge 1$, the following matrix inequalities
  hold:
$$s(A)^{2k}\le m^{k-1}n^ks(A^{(2k)}),\qquad
s(A)^{2k+1}\le m^kn^ks(A^{(2k+1)}).$$
\end{thm}


For the special case of symmetric matrices, this theorem was proved in
1959 by Mulholland and Smith~\cite{ms59}, thus settling an earlier
conjecture of Mandel and Hughes~\cite{mh58} that had been based on the
study of certain genetical models.  For arbitrary matrices (with
nonnegative entries), Theorem~\ref{thm1} also generalises the matrix
inequality
$$s(A)^3\le mns(AA^tA),$$
which was first proved in 1960 by Atkinson, Moran and
Watterson~\cite{awm60} using methods of perturbation theory.

Theorem~\ref{thm1} has a graph theoretic
interpretation when applied to matrices with entries in $\{0,1\}$. Let
$G$ be a graph with red vertices labeled $1,\dots,m$ and blue vertices
labeled $1,\dots,n$ such that every edge connects only vertices of
distinct colours: $G$ is a bipartite graph. Its reduced incidence
matrix is an $m\times n$ matrix $A$ such that $a_{i,j}=1$ if red
vertex $i$ is adjacent to blue vertex $j$, and $a_{i,j}=0$ otherwise.
Then $s(A)$ is the size of $G$, while $s(A^{(\ell)})$ is the number of
walks on $G$ of length $\ell$ starting from a red vertex, i.e., the
number of sequences $(v_0,\dots,v_\ell)$ such that $v_0$ is a red
vertex and every pair $\{v_i,v_{i+1}\}$ is an edge in $G$.
Theorem~\ref{thm1} then yields the optimal lower bound of the number
of walks in terms of the size of $G$. We do not know of a
corresponding lower bound for the number of trails (walks with no edge
repeated) or paths (walks with no vertex repeated).



Recall that an $m\times n$ matrix $A$ is said to be
\emph{bistochastic} if every row sum of $A$ is equal to $s(A)/m$, and
every column sum of $A$ is equal to $s(A)/n$.  In
Section~\ref{sec:asymptotic} we prove the following asymptotic form of
Theorem~\ref{thm1}:

\begin{thm}
  \label{thm2}
  Let $A$ be an $m\times n$ matrix with nonnegative real entries.  If
  $A$ is bistochastic, then for all $k\ge 1$,
$$s(A)^{2k}=m^{k-1}n^ks(A^{(2k)}),\qquad
s(A)^{2k+1}=m^kn^ks(A^{(2k+1)}).$$ If $A$ is not bistochastic, then
there exist constants $c>0$ and $\gamma>1$ $($depending only on $A)$
such that for all $\ell\ge 1$,
$$s(A)^\ell<c\gamma^{-\ell}(mn)^{\ell/2}s(A^{(\ell)}).$$
\end{thm}


As we show in Sections~\ref{sec:matrix} and~\ref{sec:asymptotic}, both
of the above theorems, though stated for arbitrary rectangular
matrices with nonnegative entries, follow from the special case of
\emph{square} matrices.

Theorem~\ref{thm2} has an immediate application. Atkinson,
Moran and Watterson~\cite{awm60} conjectured that for a nonnegative
symmetric kernel function $K(x,y)$ that is integrable (in a suitable
sense) over the square $0\le x,y\le a$, the inequality
\begin{equation}
  \label{eq:kernel} 
  \int\limits_0^a\int\limits_0^a K_\ell(x,y)\,\mathrm dx\,\mathrm dy\ge\frac 1{a^{\ell-1}}
  \biggl(\int\limits_0^a\int\limits_0^a K(x,y)\,\mathrm dx\,\mathrm dy\biggl)^\ell 
\end{equation}
holds for all $\ell\ge 1$.  Here $K_\ell(x,y)$ denotes the
\emph{$\ell$-th order iterate of $K(x,y)$}, which is defined
recursively by
$$K_1(x,y)=K(x,y),\qquad
K_\ell(x,y)=\int\limits_0^a K_{\ell-1}(x,t)K(t,y)\,\mathrm dt.$$


Beesack~\cite{be69} showed that the Atkinson-Moran-Watterson
conjecture follows from the matrix identities of Mulholland and Smith
described above. Using Beesack's ideas together with Theorem~\ref{thm2}, we
prove in Section~\ref{sec:integral} the following asymptotic form of
the Atkinson-Moran-Watterson inequality \eqref{eq:kernel}:

\begin{thm}
  \label{thm3}
  Let $K(x,y)$ be a nonnegative
    symmetric kernel function that is integrable over the square $0\le
    x,y\le a$, and consider the function $f(x) =\int\limits_0^a
    K(x,y)\,\mathrm dy$ defined on the interval $0\le x\le a$.  If $f(x)$ is
    constant almost everywhere, then for all $\ell\ge 1$
$$\int\limits_0^a\int\limits_0^a K_\ell(x,y)\,\mathrm dx\,\mathrm dy=
\frac 1{a^{\ell-1}}
\biggl(\int\limits_0^a\int\limits_0^a K(x,y)\,\mathrm dx\,\mathrm dy\biggl)^\ell.$$
If not,   there exist constants $c>0$ and $\gamma>1$
$($depending only on $K)$ such that for all $\ell\ge 1$
$$\int\limits_0^a\int\limits_0^a K_\ell(x,y)\,\mathrm dx\,\mathrm dy>
\frac{c\gamma^\ell}{a^{\ell-1}}
\biggl(\int\limits_0^a\int\limits_0^a K(x,y)\,\mathrm dx\,\mathrm dy\biggl)^\ell.$$
\end{thm}

\begin{rem}
  \label{noyit}
  Using an approximation argument as in the proof of
  Theorem~\ref{thm3}, Theorem~\ref{thm1} can be also applied to
  establish an analogue to inequalities \eqref{eq:kernel} and
  Theorem~\ref{thm3} in the case of nonsymmetric kernel functions.
  Let $K(x,y)$ be any nonnegative kernel function that is integrable
  on the rectangle $0\le x\le a$, $0\le y\le b$ and let $K_\ell$ be
  the $\ell$-th order iterate of $K$ defined by $K_1(x,y)=K(x,y)$ and
  for each integer $k \geq 1$,
  \begin{align*}
    K_{2k}(x,x')&=\int\limits_0^bK_{2k-1}(x,y)K(x',y)\,\mathrm dy,\\
    K_{2k+1}(x,y)&=\int\limits_0^aK_{2k}(x,x')K(x',y)\,\mathrm dx'.
  \end{align*}
  In this case, inequalities \eqref{eq:kernel} become
$$
\int\limits_0^a\int\limits_0^b K_{2k+1}(x,y)\,\mathrm dx\,\mathrm dy
\geq\frac{1}{a^k b^k} \biggl(\int\limits_0^a\int\limits_0^b
K(x,y)\,\mathrm dx\,\mathrm dy\biggl)^{2k+1}$$
$$\int\limits_0^a\int\limits_0^a K_{2k}(x,x')\,\mathrm dx\,\mathrm dx'
\geq\frac{1}{a^{k-1}b^k} \biggl(\int\limits_0^a\int\limits_0^b
K(x,y)\,\mathrm dx\,\mathrm dy\biggl)^{2k}.$$ The analogue of
Theorem~\ref{thm3} is then obvious.
\end{rem}

%%%%%%%%%%%%%%%%%%%%%
% MATRIX INEQUALITY %
%%%%%%%%%%%%%%%%%%%%%

\section{Matrix inequality}
\label{sec:matrix}

\noindent Given a matrix $A=(a_{i,j})$ and an integer $\ell\ge 0$,
we denote by $a_{i,j}^{(\ell)}$ the $(i,j)$-th entry of $A^{(\ell)}$,
so that $A^{(\ell)}=(a_{i,j}^{(\ell)})$.
This notation will be used often in the sequel.


\begin{lem}
Let $B=(b_{i,j})$ be a $d\times d$ matrix with
    nonnegative real entries.  For any two sequences $\{\alpha_i\}$
    and $\{\beta_i\}$ of nonnegative real numbers, the following
    inequality holds:
$$(I'_2):\quad\qquad\sum\limits_{i,j=1}^d \alpha_i\beta_ib_{i,j}
\le d^{\frac{1}{ 2}} \biggl(\sum\limits_{i,j=1}^d
\alpha_i^2\beta_j^2b_{i,j}^{(2)} \biggl)^{\frac{1}{2}}.$$
\end{lem}

\begin{proof}
  To prove the lemma, we apply the Cauchy-Schwarz inequality twice as
  follows:
  \begin{eqnarray} \label{eq:lemma} \sum\limits_{i,j=1}^d
    \alpha_i\beta_ib_{i,j} =\sum\limits_{i,k=1}^d
    \alpha_i\beta_ib_{i,k} \le d^{\frac{1}{2}}
    \biggl(\sum\limits_{k=1}^d \Big(\sum\limits_{i=1}^d
    \alpha_i\beta_ib_{i,k}\Big)^2\biggl)^{\frac{1}{2}}.
  \end{eqnarray}
  \begin{displaymath} \begin{array}{ll} \sum\limits_{i,j=1}^d
      \alpha_i\beta_ib_{i,j} & \leq d^{\frac{1}{2}}
      \biggl(\sum\limits_{i,j,k=1}^d
      \alpha_i\alpha_j\beta_i\beta_j
      b_{i,k}b_{j,k}\biggl)^{\frac{1}{2}}\cr &= d^{\frac{1}{2}}
      \biggl(\sum\limits_{i,j=1}^d
      \alpha_i\alpha_j\beta_i\beta_j
      b_{i,j}^{(2)}\biggl)^{\frac{1}{2}}\cr &= d^{\frac{1}{2}}
      \biggl(\sum\limits_{i,j=1}^d
      \alpha_i\beta_j(b_{i,j}^{(2)})^{\frac{1}{2}}
      \cdot\alpha_j\beta_i(b_{j,i}^{(2)})^{\frac{1}{2}}\biggl)^{\frac{1}{2}}\cr
      &\le d^{\frac{1}{2}} \biggl(\sum\limits_{i,j=1}^d
      \alpha_i^2\beta_j^2b_{i,j}^{(2)} \biggl)^{\frac{1}{2}}.
    \end{array}
  \end{displaymath}
  Here we have used the fact that $B^{(2)}=BB^t$ is a symmetric
  matrix.
\end{proof}

\begin{thm}
  \label{thm1bis}
  Let $B=(b_{i,j})$ be a
    square $d\times d$ matrix with nonnegative real entries, and let
    $\{\alpha_i\}$ be any sequence of nonnegative real numbers.  Then
    for each integer $\ell\ge 1$, we have
$$(I_\ell):\qquad\qquad 
\sum\limits_{i,j=1}^d \alpha_ib_{i,j} \le d^{\frac{\ell-1}\ell}
\biggl(\sum\limits_{i,j=1}^d \alpha_i^\ell b_{i,j}^{(\ell)}
\biggl)^{\frac 1{\ell}}.$$
\end{thm}

\begin{proof}[Proof of Theorem~\ref{thm1bis}]
  The case $\ell=1$ is trivial while the case $\ell=2$ is a
  consequence of the lemma above. We prove the general case by
  induction.  Suppose that $p\ge 2$, and the inequalities
  $(I_1),(I_2),\ldots,(I_p)$ hold for all square matrices with
  nonnegative real entries.  If $p=2k-1$ is an odd integer, then the
  inequality $(I_{p+1})$ follows immediately from $(I_2)$ and $(I_k)$.
  Indeed, since $B^{(2k)}=B^{(2)(k)}$, we have
  \begin{equation}
    \label{eq:even_ineq}
    \sum\limits_{i,j=1}^d \alpha_ib_{i,j}
    \le d^{\frac{1}{2}} \biggl(\sum\limits_{i,j=1}^d
    \alpha_i^2b_{i,j}^{(2)}\biggl)^{\frac{1}{2}}
    \le d^{\frac{1}{2}} \biggl(d^{\frac{k-1}{k}}\Big(
    \sum\limits_{i,j=1}^d \alpha_i^{2k}
    b_{i,j}^{(2)(k)}\Big)^{\frac{1}{k}}\biggl)^{\frac{1}{2}}. 
  \end{equation}
  Thus
  \begin{displaymath}
    \sum\limits_{i,j=1}^d \alpha_ib_{i,j}
    \le d^{\frac{2k-1}{2k}}\biggl(
    \sum\limits_{i,j=1}^d \alpha_i^{2k}b_{i,j}^{(2k)}\biggl)^{\frac{1}{2k}}. 
  \end{displaymath}



If $p=2k$ is an even integer, then the inequality $(I_{p+1})$ follows
from H\"older's inequality, and the inequalities $(I_k)$ and $(I'_2)$.
Indeed, by H\"older's inequality, we have
\begin{equation}
  \label{eq:odd_ineq}
  \sum\limits_{i,j=1}^d \alpha_ib_{i,j}
  \le d^{\frac{1}{2k+1}} \biggl(\sum\limits_{i=1}^d\alpha_i^{\frac{2k+1}{2k}}
  \Big(\sum\limits_{j=1}^d
  b_{i,j}\Big)^{\frac{2k+1}{2k}}\biggl)^{\frac{2k}{2k+1}}.
\end{equation}
Let ${\mathcal I}$ denote the term between parentheses, and set
$\beta_i=\sum\limits_{j=1}^d b_{i,j}$ for each $i$.  Then
$${\mathcal I}=\sum\limits_{i=1}^d\alpha_i^{\frac{2k+1}{2k}}
\Big(\sum\limits_{j=1}^d b_{i,j}\Big)^{\frac{2k+1}{2k}}
=\sum\limits_{i,j=1}^d
\alpha_i^{\frac{2k+1}{2k}}\beta_i^{\frac{1}{2k}}b_{i,j}.$$
Applying $(I_k)$, it follows that
$${\mathcal I}\le d^{\frac{k-1}{k}}
\biggl(\sum\limits_{i,j=1}^d
\alpha_i^{\frac{2k+1}{2}}\beta_i^{\frac{1}{2}}
b_{i,j}^{(k)}\biggl)^{\frac{1}{k}}.$$ Applying the lemma to the
sequences $\{\alpha_i^{\frac{2k+1}{2}}\}$ and
$\{\beta_i^{\frac{1}{2}}\}$, and using the fact that
$B^{(k)(2)}=B^{(2k)}$, we see that
\begin{displaymath} {\mathcal I}
  \le d^{\frac{k-1}{k}}
  \biggl(d^{\frac{1}{2}}\Big(\sum\limits_{i,j=1}^d
  \alpha_i^{2k+1}\beta_j
  b_{i,j}^{(k)(2)}\Big)^{\frac{1}{2}}\biggl)^{\frac{1}{k}}
  =d^{\frac{2k-1}{2k}}
  \biggl(\sum\limits_{i,j=1}^d
  \alpha_i^{2k+1}\beta_jb_{i,j}^{(2k)}\biggl)^{\frac{1}{2k}}. 
\end{displaymath}
Putting everything together, we have therefore shown that
$$\sum\limits_{i,j=1}^d \alpha_ib_{i,j}
\le d^{\frac{2k}{2k+1}} \biggl(\sum\limits_{i,j=1}^d
\alpha_i^{2k+1}\beta_jb_{i,j}^{(2k)}\biggl)^{\frac{1}{2k+1}}.$$
Finally, note that
$$\sum\limits_{j=1}^d\beta_jb_{i,j}^{(2k)}
=\sum\limits_{\ell=1}^d b_{i,\ell}^{(2k)}\beta_\ell
=\sum\limits_{j,\ell=1}^d b_{i,\ell}^{(2k)}b_{\ell,j}
=\sum\limits_{j=1}^d b_{i,j}^{(2k+1)}$$ since $B^{(2k+1)}=B^{(2k)}B$.
Consequently,
\begin{equation}
  \label{eq:odd_ineq2}
  \sum\limits_{i,j=1}^d \alpha_ib_{i,j}
  \le d^{\frac{2k}{2k+1}}
  \biggl(\sum\limits_{i,j=1}^d
  \alpha_i^{2k+1}b_{i,j}^{(2k+1)}\biggl)^{\frac{1}{2k+1}}
\end{equation}
and $(I_{p+1})$ holds for the case $p=2k$.  Theorem~\ref{thm1bis} now follows
by induction.
\end{proof}

 
\begin{proof}[Proof of Theorem~\ref{thm1}]
  For the case of square matrices, Theorem~\ref{thm1} follows immediately from
  Theorem~\ref{thm1bis}.  Indeed, taking $\alpha_i=1$ for each $i$, the
  inequality $(I_\ell)$ yields the corresponding inequality in
  Theorem~\ref{thm1}.


Now, let $A$ be an $m\times n$ matrix with nonnegative real entries,
put $d=mn$, and let $B$ be the $d\times d$ matrix with nonnegative
real entries defined as the tensor product $B=A\otimes J_{n,m}$,
where $J_{n,m}$ is the $n\times m$ matrix with every entry equal to
1.  For any integers $\ell,k \ge 0$, the relations
\begin{displaymath}
  B^{(\ell)}=A^{(\ell)}\otimes J_{n,m}^{(\ell)}, \quad
  s\big(B^{(\ell)}\big)=s\big(A^{(\ell)}\big)s(J_{n,m}^{(\ell)}),
\end{displaymath}
\begin{displaymath}
  s\big(J_{n,m}^{(2k)}\big)=m^kn^{k+1}, \quad
  s\big(J_{n,m}^{(2k+1)}\big)=m^{k+1}n^{k+1}.
\end{displaymath}
are easily checked. In particular, $s(B)=mns(A)$.  Applying
Theorem~\ref{thm1} to the matrix $B$ and using these identities, the
inequalities of Theorem~\ref{thm1} follow for the matrix $A$.
\end{proof}


 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ASYMPTOTIC MATRIX INEQUALITY %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Asymptotic matrix inequality}
\label{sec:asymptotic}

\noindent As will be shown below, Theorem~\ref{thm2} is a consequence of the following
more precise theorem for square matrices:

\begin{thm}
  \label{thm2bis}
  Let $B$ be a square $d\times
    d$ matrix with nonnegative real entries and $s(B)\ne 0$.  Let
    $\lambda$ be the largest eigenvalue of $B^{(2)}=BB^t$, and put
    $\gamma=\lambda d^2\slash s(B)^2$.  Then $\gamma\ge 1$, and there
    exists a constant $c>0$ $($depending only on $B)$ such that for
    all integers $\ell\ge 0$,
    \begin{eqnarray}
\label{eq:inequality} 
s(B)^\ell<c\gamma^{-{\frac\ell 2}}d^{\ell-1}s(B^{(\ell)}).
\end{eqnarray}
Moreover, the following assertions are equivalent:
{\renewcommand{\labelenumi}{$(\alph{enumi})$}
\begin{enumerate}
\item $\gamma=1$,
\item $s(B)^\ell=d^{\ell-1}s(B^{(\ell)})$ for every integer $\ell\ge
  0$,
\item $s(B)^\ell= d^{\ell-1}s(B^{(\ell)})$ for some integer $\ell\ge
  3$,
\item $B$ is bistochastic.
\end{enumerate}}
\end{thm}

\begin{proof}
  We express $B^{(2)}=BB^t$ in the form
  $B^{(2)}=U^tDU$, where $U=(u_{i,j})$ is an orthogonal matrix, and
  $D$ is a diagonal matrix $\diag(\lambda_1,\ldots,\lambda_d)$ with
  $\lambda_1\ge\ldots\ge\lambda_d\ge 0$.  Here $\lambda=\lambda_1$.
  For each $\nu=1,\ldots,d$, let $E_\nu$ be the projection matrix
  whose $(\nu,\nu)$-th entry is 1, and all other entries are equal
  to 0.  Put $A_\nu=U^tE_\nu U$ for each $\nu$.  Then for all
  integers $k\ge 0$,
$$B^{(2k)}=\sum\limits_{\nu=1}^d\lambda_\nu^kA_\nu,\qquad
B^{(2k+1)}=\sum\limits_{\nu=1}^d\lambda_\nu^kA_\nu B.$$ By a
straightforward calculation, we see that for each $\nu$
\begin{equation}
  \label{eq:sAnu}
  s(A_\nu)=\Big(\sum\limits_{i=1}^d u_{\nu,i}\Big)^2,\qquad
  s(A_\nu B)=\biggl(\sum\limits_{i=1}^d
  u_{\nu,i}\biggl)\Big(\sum\limits_{j,k=1}^d u_{\nu,k}b_{k,j}\Big).
\end{equation}
In particular, $s(A_\nu)\ge 0$.  By Theorem~\ref{thm1bis}, it follows that
\begin{equation}
  \label{eq:gamma} 
  \frac{s(B)^2}d \le s(B^{(2)})=\sum\limits_{\nu=1}^d\lambda_\nu s(A_\nu)
  \le\lambda\sum\limits_{\nu=1}^ds(A_\nu)  =\lambda d.
\end{equation}
Therefore, $\gamma=\displaystyle{\frac{\lambda d^2}{s(B)^2}}\ge 1.$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now, from the definition of $\gamma$, we have  
$$\frac{\gamma^{\frac{\ell}{2}} s(B)^\ell} {d^{\ell -1} s(B^{(\ell)})}
= d \frac{\lambda^{\frac{\ell}{2}}}{s(B^{(\ell)})}\cdot$$ Then, in
order to show inequality \eqref{eq:inequality}, we will show that the
${\lambda^{\frac{\ell}{2}}}\slash{s(B^{(\ell)})}$ are bounded above by
a constant that is independent of $\ell$.  Indeed, let $C_\ell=
{B^{(\ell)}}\slash{s(B^{(\ell)})}$ for every $\ell\ge 0$.  Since each
$C_\ell$ has nonnegative real entries, and $s(C_\ell)=1$, the entries
of $C_\ell$ all lie in the closed interval $[0,1]$.  Thus the entries
of the matrices $UC_{2k}U^t$ and $UC_{2k+1}B^tU^t$ are bounded by a
constant that depends only on $B$.  Noting that for each nonnegative
integer $k$, we have $$UC_{2k}U^t=\frac{D^k}{s(B^{(2k)})}, \qquad
UC_{2k+1}B^tU^t =\frac{D^{k+1}}{s(B^{(2k+1)})},$$ and on examining the
$(1,1)$th entry for each of these matrices, we see that
${\lambda^k}\slash{s(B^{(2k)})}$ and
${\lambda^{k+1}}\slash{s(B^{(2k+1)})}$ are both bounded above by a
constant that is independent of $k$.  Consequently, inequality
\eqref{eq:inequality} holds.


 


$(a){\Rightarrow}(b)$: 
If $\gamma=1$, then $\lambda d=s(B)^2/d$, hence from
\eqref{eq:gamma} 
we see that $s(A_\nu)=0$ whenever $\lambda_\nu\ne\lambda$.  
By \eqref{eq:sAnu}, we also have that
$s(A_\nu B)=0$ whenever $\lambda_\nu\ne\lambda$.  Thus
\begin{equation*}
  \begin{split}
    s(B^{(2k)})&=\sum\limits_{\nu=1}^d\lambda_\nu^ks(A_\nu)
    =\lambda^k\sum\limits_{\nu:\lambda_\nu=\lambda}s(A_\nu)\\
    &=\lambda^k\sum\limits_{\nu=1}^d s(A_\nu)=\lambda^kd =
    \displaystyle{\frac{s(B)^{2k}}{d^{2k-1}}},
\\
  s(B^{(2k+1)})&=\sum\limits_{\nu=1}^d\lambda_\nu^ks(A_\nu B)
  =\lambda^k\sum\limits_{\nu:\lambda_\nu=\lambda}
   s(A_\nu B) \\
  &=\lambda^k\sum\limits_{\nu=1}^d s(A_\nu
  B)=\lambda^ks(B)= \displaystyle{\frac{s(B)^{2k+1}}{d^{2k}}}\cdot
\end{split}
\end{equation*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$(b){\Rightarrow}(a)$:
If $(b)$ holds, then inequality \eqref{eq:inequality} 
 implies $1< c \gamma^{-\frac{\ell}{2}}$ for some $\gamma \geq 1$
and all integers $\ell \geq 0$. This forces $\gamma = 1$. 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$(b){\Rightarrow}(c)$: Trivial.

$(c){\Rightarrow}(d)$: 
Suppose that $\ell=2k+1\ge 3$ is an odd integer such that
$s(B)^\ell= d^{\ell-1}s(B^{(\ell)})$.  Taking every $\alpha_i=1$ in
the proof of Theorem~\ref{thm1bis}, our hypothesis means that equality holds
in~\eqref{eq:odd_ineq2}, hence~\eqref{eq:odd_ineq} must also hold with
equality:
$$
\sum\limits_{i,j=1}^d b_{i,j}
=d^{\frac{1}{2k+1}} \biggl(\sum\limits_{i=1}^d
\Big(\sum\limits_{j=1}^d
b_{i,j}\Big)^{\frac{2k+1}{2k}}\biggl)^{\frac{2k}{2k+1}}.
$$
By H\"older's inequality, this is only possible if all of the
row sums of $B$ are equal.  Since $\ell$ is odd and $s$ is
transpose-invariant, we also have
$$s(B^t)^\ell= d^{\ell-1}s\big((B^{(\ell)})^t\big)
= d^{\ell-1}s\big((B^t)^{(\ell)}\big).$$
Thus all of the row sums of $B^t$ are equal as well,
and $B$ is bistochastic.

 
Now suppose that $\ell=2k\ge 4$ is an even integer  such that
$s(B)^\ell= d^{\ell-1}\*s(B^{(\ell)})$.  By taking
every $\alpha_i=1$ in~\eqref{eq:even_ineq}, we see that
$s(B)^2= ds(B^{(2)})$.  Then, taking every $\alpha_i=\beta_i=1$  
in the proof of the lemma, we see that equality holds
in~\eqref{eq:lemma} which is  only
possible if all of the
column sums of $B$ are equal.  
Therefore $s(BA)=\beta s(A)$ for every $d\times d$ matrix $A$, where
$\beta = s(B)/d$ is the sum of each column of $B$. 
In particular,
\begin{equation*}
  \begin{split}
    s(B)^\ell&=d^{\ell-1}s\big(B^{(\ell)}\big)
    =d^{\ell-1}\beta s\big((B^t)^{(\ell-1)}\big)\\
    &=d^{\ell-1}\beta s\big((B^{(\ell-1)})^t\big)
    =d^{\ell-1}\beta s(B^{(\ell-1)}),
\end{split}
\end{equation*}
thus $s(B)^{\ell-1}= d^{\ell-2}s(B^{(\ell-1)})$.  Since
$\ell -1$ is odd, we can apply the previous result to conclude
that $B$ is bistochastic.

$(d){\Rightarrow}(b)$: 
Suppose $B$ is bistochastic, with every row or
column sum equal to $\beta=s(B)/d$.  For any $d \times d$ matrix $A$, 
one has $s(AB)=\beta s(A)$ and $s(AB^t)=\beta s(A)$.  In particular, 
$s(B^{(2k+1)})=\beta s(B^{(2k)})$ and 
$s(B^{(2k+2)}) =\beta s(B^{(2k+1)})$  for all $k\ge 0$.
Consequently,
$$s(B^{(\ell)})=\beta^{\ell-1}s(B)=\frac{s(B)^\ell}{d^{\ell-1}},\qquad
\ell\ge 0.$$
This completes the proof.
\end{proof}

\begin{cor}
  Let $B$ be a square $d\times d$ matrix with nonnegative real entries
  and $s(B)\ne 0$.  Let $\beta_j$ be the $j$-th column sum of $B$ for
  each $j$, and put
$$\delta=1+\frac 1{2s(B)^2}
\sum\limits_{i,j=1}^d(\beta_i-\beta_j)^2.$$ Then there exists a
constant $c>0$ $($depending only on $B)$ such that for all $\ell\ge
0$, we have
$$s(B)^\ell<c\delta^{-{\frac\ell 2}}d^{\ell-1}s(B^{(\ell)}).$$
\end{cor}

\begin{proof}
  Note first that for \emph{any} $d\times d$ matrix $B$, if $\beta_j$
  denotes the $j$-th column sum of $B$, then it is easily seen that
  \begin{equation}
    \label{eq:equality}
    s(B^{(2)})=\frac{s(B)^2}d+\frac 1{2d}\sum\limits_{i,j=1}^d
    (\beta_i-\beta_j)^2.
  \end{equation}
  Using the notation of Theorem~\ref{thm2bis} and applying the relations
  \eqref{eq:gamma} and \eqref{eq:equality}, we have
$$\gamma=\frac{\lambda d^2}{s(B)^2}\ge
\frac{ds(B^{(2)})}{s(B)^2}= 1+\frac 1{2s(B)^2}
\sum\limits_{i,j=1}^d(\beta_i-\beta_j)^2=\delta.$$ The corollary
therefore follows from \eqref{eq:inequality}.
\end{proof}



\begin{proof}[Proof of Theorem~\ref{thm2}]
  Given an $m\times n$ matrix $A$ with nonnegative real entries, we
  proceed as in the proof of Theorem~\ref{thm1}: put $d=mn$, and let
  $B=A\otimes J_{n,m}$.  Note that $A$ is bistochastic if and only if
  $B$ is bistochastic.  Applying the corollary above to $B$, Theorem~\ref{thm2}
  follows immediately for the matrix $A$. The details are left to the
  reader.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ASYMPTOTIC INTEGRAL INEQUALITY %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Asymptotic kernel inequality}
\label{sec:integral}

\begin{proof}[Proof of Theorem~\ref{thm3}]
  By changing variables if necessary, we can assume that $a=1$.  For
  simplicity, we will also assume that $K(x,y)$ is continuous.
  Consider the function $f(x)$ defined by
$$f(x)=\int\limits_0^1 K(x,y)\,\mathrm dy,\qquad x\in[0,1].$$

If $f(x)$ is a constant function, then since
$K(x,y)$ is symmetric, the equality
$$\int\limits_0^1 \int\limits_0^1 K_\ell(x,y)\,\mathrm dx\,\mathrm dy= 
\biggl(\int\limits_0^1 \int\limits_0^1 K(x,y)\,\mathrm dx\,\mathrm dy\biggl)^\ell$$
for all $\ell\ge 1$ follows from an easy inductive argument.


Now suppose that $f(x)$ is not constant, and let $m$
and $M$ denote respectively the minimum and maximum value of $f(x)$ on
$[0,1]$.  Choose $\varepsilon>0$ such that $4\varepsilon<M-m$.  For
every integer $d\ge 1$, let $\UU_i^{[d]}$ be the open interval
$$\UU_i^{[d]}=\Big(\frac{i-1}{d},\frac{i}{d}\Big),\qquad 1\le i\le d,$$
and let $\UU_{i,j}^{[d]}$ be the rectangle $\UU_i^{[d]}\times\UU_j^{[d]}$
for $1\le i,j\le d$.  Let $K^{[d]}(x,y)$ be defined on
$[0,1]\times[0,1]$ as follows:
\begin{displaymath}
  \begin{cases}
    \min\Big\{K(s,t):(s,t)\in\overline{\UU_{i,j}^{[d]}}\Big\}
    &\text{if $(x,y)\in\UU_{i,j}^{[d]}$ for some $1\le i,j\le d$}\\
    K(x,y)&\text{otherwise.}
  \end{cases}
\end{displaymath}
Here $\overline{\UU_{i,j}^{[d]}}$ denotes the closure of
$\UU_{i,j}^{[d]}$.  Noting that $K^{[d]}(x,y)$ is constant on each
rectangle $\UU_{i,j}^{[d]}$, let $B_{[d]}$ be the $d\times d$ matrix
whose $(i,j)$-th entry is equal to $K^{[d]}(\UU_{i,j}^{[d]})$.  Let
$K_\ell^{[d]}(x,y)$ denote the $\ell$-th order iterate of
$K^{[d]}(x,y)$ for each $\ell\ge 1$.  Then
$$K_\ell^{[d]}(x,y)=
\int\limits_0^1 K_{\ell-1}^{[d]}(x,t) K^{[d]}(t,y)\,\mathrm dt=
\sum\limits_{k=1}^{d}\int\limits_{\UU_k^{[d]}}
K_{\ell-1}^{[d]}(x,t)K^{[d]}(t,y)\,\mathrm dt.$$ It follows by induction
that $K_\ell^{[d]}(x,y)$ is also constant on each rectangle
$\UU_{i,j}^{[d]}$, and
$$K_\ell^{[d]}(\UU_{i,j}^{[d]})=\frac 1 d\sum\limits_{k=1}^{d}
K_{\ell-1}^{[d]}(\UU_{i,k}^{[d]})K^{[d]}(\UU_{k,j}^{[d]});$$ by
induction, this is the $(i,j)$-th entry of the matrix
$\displaystyle{\frac 1{d^{\ell-1}} B_{[d]}^{(\ell)}}$.  In other
words,
\begin{equation}
  \label{eq:relation}
  \Big(K_\ell^{[d]}(\UU_{i,j}^{[d]})\Big)=
  \frac 1{d^{\ell-1}}B_{[d]}^{(\ell)},\qquad\text{for all }\ell,d\ge 1.
\end{equation}
Now since $f(x)$ is continuous, we can choose $d$ sufficiently large
such that for some integers $1\le i_m,i_M\le d$, we have
$$f(x)< m +\varepsilon,\qquad\text{for all }x \in \UU_{i_m}^{[d]},$$
$$f(x)> M -\varepsilon,\qquad\text{for all }x \in \UU_{i_M}^{[d]}.$$
Taking $d$ larger if necessary, we can further assume that
$$0\le K(x,y)-K^{[d]}(x,y)<\varepsilon$$ for all 
$0\le x,y\le 1$. Fixing this value of $d$, we define
$$\gamma=1+\frac{\varepsilon^2}{2d^2
  \Big(\int\limits_0^1\int\limits_0^1 K(x,y)\,\mathrm dx\,\mathrm dy\Big)^2}.$$
Finally, since $\gamma^{-{\frac 1 4}}<1$, we can choose $e$
sufficiently large so that $K^{[de]}(x,y)>\gamma^{-{\frac 1
    4}}K(x,y)$ for all $0\le x,y\le 1$.  For this value of $e$, we
therefore have
$$\int\limits_0^1\int\limits_0^1 K^{[de]}(x,y)\,\mathrm dx\,\mathrm dy
>\gamma^{-{\frac 1 4}}\int\limits_0^1\int\limits_0^1 K(x,y)\,\mathrm dx\,\mathrm dy.$$

By the corollary to Theorem~\ref{thm2bis} applied to the
matrix $B_{[de]}$, there exists a constant $c>0$, which is independent
of $\ell$, such that
$$s\big(B_{[de]}\big)^\ell<c
\delta^{-{\frac\ell 2}}(de)^{\ell-1}s\big(B_{[de]}^{(\ell)}\big)$$
for all integers $\ell\ge 0$, where
$$\delta=1+\frac 1{2s\big(B_{[de]}\big)^2}
\sum\limits_{i,j=1}^{de}\big(\beta_{[de],i}-\beta_{[de],j}\big)^2.$$
Here $\beta_{[de],j}$ denotes the $j$-th column sum of $B_{[de]}$ for
each $j$.  We now claim that $\delta>\gamma$.  Granting this fact for
the moment, we apply \eqref{eq:relation} to $K^{[de]}(x,y)$ and
obtain:
$$\displaylines{
  \int\limits_0^1\int\limits_0^1 K_\ell(x,y)\,\mathrm dx\,\mathrm dy
  \ge\int\limits_0^1\int\limits_0^1 K_\ell^{[de]}(x,y)\,\mathrm dx\,\mathrm dy ={\frac
    1{(de)^2}}\sum\limits_{i,j=1}^{de}
  K_\ell^{[de]}\Big(\UU_{i,j}^{[de]}\Big) \cr
  =\frac{1}{(de)^{\ell+1}}{s\big(B_{[de]}^{(\ell)}\big)}
  >c^{-1}\delta^{\frac\ell
    2}(de)^{-2\ell}s\big(B_{[de]}\big)^\ell\cr
  =c^{-1}\delta^{\frac\ell 2}\biggl({\frac 1{(de)^2}}
  \sum\limits_{i,j=1}^{de}K^{[de]}\Big(\UU_{i,j}^{[de]}\Big)\biggl)^\ell
  =c^{-1}\delta^{\frac\ell 2}\biggl(\int\limits_0^1\int\limits_0^1
  K^{[de]}(x,y)\,\mathrm dx\,\mathrm dy\biggl)^\ell\cr >c^{-1}\delta^{\frac\ell
    2}\gamma^{-{\frac\ell 4}} \biggl(\int\limits_0^1\int\limits_0^1
  K(x,y)\,\mathrm dx\,\mathrm dy\biggl)^\ell > c^{-1}\gamma^{{\frac\ell
      4}}\biggl(\int\limits_0^1\int\limits_0^1
  K(x,y)\,\mathrm dx\,\mathrm dy\biggl)^\ell.}$$ This completes the proof of the
theorem modulo our claim that $\delta>\gamma$.  To see this, let $\V$
be any interval of the form $\UU_{i}^{[de]}$ such that
$\V\subset\UU_{i_m}^{[d]}$.  Note that there are $e$ such intervals.
Since $B^{[de]}$ is a symmetric matrix, the column sum
$\beta_{[de],\V}$ of $B_{[de]}$ corresponding to the interval $\V$ is
equal to the ``$\V$-th'' row sum, which can be bounded as follows:
\begin{multline*}
  \beta_{[de],\V}=
  \sum\limits_{j=1}^{de}K^{[de]}\big(\V,\UU_j^{[de]}\big)
  =(de)^2\int\limits_{\V}\int\limits_0^1 K^{[de]}(x,y)\,\mathrm dy\,\mathrm dx \\
  \le (de)^2\int\limits_{\V}\int\limits_0^1 K(x,y)\,\mathrm dy\,\mathrm dx
  =(de)^2\int\limits_{\V}f(x)\,\mathrm dx<de(m+\varepsilon).
\end{multline*}
Similarly, let $\W$ be any interval of the form $\UU_i^{[de]}$ such
that $\W\subset\UU_{i_M}^{[d]}$.  Again, there are $e$ such intervals,
and by a similar calculation, the column sum $\beta_{[de],\W}$
satisfies the bound
$$\displaylines{
  \beta_{[de],\W}=
  \sum\limits_{j=1}^{de}K^{[de]}\big(\W,\UU_j^{[de]}\big) >
  de(M-2\varepsilon).\cr }$$ Thus
$$\sum\limits_{i,j=1}^{de}\big(\beta_{[de],i}-\beta_{[de],j}\big)^2
\ge\sum\limits_{\V,\W}\big(\beta_{[de],\W}-\beta_{[de],\V}\big)^2
>d^2e^4(M-m-3\varepsilon)^2>d^2e^4\varepsilon^2.$$ On the other hand,
we have
$$\displaylines{
  s\big(B_{[de]}\big) = (de)^2\int\limits_0^1\int\limits_0^1
  K^{[de]}(x,y)\,\mathrm dx\,\mathrm dy \le(de)^2\int\limits_0^1\int\limits_0^1
  K(x,y)\,\mathrm dx\,\mathrm dy,}$$ and the claim follows.
\end{proof}

\noindent
\emph{Keywords.} Matrix inequality, iterated kernel. 

\bigskip

\noindent 
2000 \emph{Mathematics Subject Classification.} 15A45.


% REFERENCES %
%%%%%%%%%%%%%%


%\begin{thebibliography}{99}
%
%
%
%
%
%\bibitem{awm60}  
%{\sc F.V.\ Atkinson, G.A.\ Watterson and P.A.P.\ Moran}, 
%{\it A matrix inequality\/},
%Quart.\ J.\ Math.\ Oxford Ser.\ (2) {\bf 11} (1960) 137--140. 
%%
%%
%\bibitem{be69}  
%{\sc P.R.\ Beesack}, 
%{\it  Inequalities involving iterated kernels and convolutions\/},
%Univ.\ Beograd.\ Publ.\ Elektrotehn.\ Fak.\ Ser.\ Mat.\ Fiz.\ {\bf 274--301}
%(1969) 11--16.  
%%
%%
%\bibitem{mh58}  
%{\sc S.P.M.\ Mandel and I.M.\ Hughes}, 
%{\it  Change in mean viability at a multiallelic locus\/},
%Nature {\bf 182} (1958) 63--64.   
%%
%%
%\bibitem{ms59}  
%{\sc H.P.\ Mulholland and C.A.B.\ Smith}, 
%{\it An inequality arising in genetical theory\/},
%Amer.\ Math.\ Monthly {\bf 66} (1959) 673--683.
%\end{thebibliography}




%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "e:/hdr/h"
%%% End: 
\clearpage

\chapter{The size of bipartite graphs with girth eight}
\label{siz}
\begin{abstract}
  \noindent Reiman's inequality for the size of bipartite graphs of girth six is
  generalised to girth eight. It is optimal in as far as
  it admits the algebraic structure of generalised quadrangles as case
  of equality. This enables us to obtain the optimal estimate $e\sim
  v^{4/3}$ for balanced bipartite graphs. We also get an optimal
  estimate for very unbalanced graphs.
\end{abstract}

\section{Introduction}

\noindent De Caen and Sz\'ekely recently proposed a new bound for the size of a 
bipartite graph of girth eight, that is a bipartite graph without cycle of
length four and six. We adapt their method to obtain the following cubic
inequality.
%
\begin{thm}\label{TTHHMM}
  Let $G$ be a bipartite graph on $v+w$ vertices. 
  \begin{itemize}
  \item[$(i)$]If $G$ contains no cycle of length 4 and 6, then its
    size $e$ satisfies
$$
e^3-(v+w)e^2+2vwe-v^2w^2\le 0.$$
  \item[$(ii)$]If $v\ge\lfloor w^2/4\rfloor$, then furthermore $e\le
  v+\lfloor w^2/4\rfloor$.
\end{itemize}
\end{thm}
%
Part $(i)$ is the right generalisation of Reiman's inequality for
bipartite graphs of girth 6 (see Prop.~\ref{prp6}) to girth 8. It
is optimal in the sense that it is an equality for all known extremal
graphs constructed via finite fields. Part $(ii)$ describes the case
of very unbalanced bipartite graphs and is optimal: there is a graph,
constructed by hand, for which it is an equality.

Let us give a brief description of this article. 
Section 2 describes a way to translate uncoloured graphs into
bipartite graphs and its converse. This permits to get two
propositions on very unbalanced graphs.

Section 3 summarises facts about bipartite graphs of girth six that
should be folklore and well known although I did not see them printed.

Section 4 is the core of the paper. We adapt an inequality of
Atkinson et al.\ to get an optimal lower bound on the number of
paths of length 3 in a bipartite graph
(Cor.~\ref{path3}). This enables us to bypass the final step in the proof
of \cite[Th.~1]{ds97} and to get our theorem.












\section{Uncoloured graphs and bipartite graphs}
\label{sec:c}



























\subsection{Expanding a graph to a bipartite graph}
\label{sec:c1}









\noindent We propose the following construction of a bipartite graph out of an
uncoloured graph. Let $G'$ be an uncoloured graph with set of vertices
$V$. Then the bipartite graph $G$ is defined as follows:

\begin{itemize}
\item the first class of vertices of $G$ is $V$;
\item the second class $W$ of vertices of $G$ is the set of
  edges of $G'$;
\item the set of edges of $G$ is $\bigl\{\,\{x,y\}:y\text{ is an
    edge of $G'$ with endpoint }x\,\bigr\}.$
\end{itemize}

Thus every vertex of $W$ has degree 2 and the size
of $G$ is twice the size of $G'$.



\subsection{Contracting a bipartite graph to an uncoloured graph}
\label{sec:c2}

\noindent Let us describe an inverse construction. 
Let $G$ be a bipartite graph with colour classes $V$ and $W$. 
Let $G'$ be the following graph: 
\begin{itemize}
\item its set of vertices is $V$;
\item its set of edges is 
$\bigl\{\,\{x,z\}\subseteq V:\exists
y~\{x,y\}~\text{and}~\{z,y\}~\text{are edges of}~G\,\bigr\}.$
\end{itemize}

The size of $G'$ is at most half the size of $G$. 
If $G$ contains no cycle of length 4, then, given $\{x,z\}$, there
is at most one $y$ such that $\{x,y\}$ and $\{z,y\}$ are edges of $G$,
so that the size of $G'$ is exactly
\begin{equation}\label{sd}
\sum_{y\in W}\binom{d(y)}2\le\binom{\#V}2.
\end{equation}
(We
recognise here \cite[Inequality $(2)$, p.~310]{bo78} for $s=t=2$.) 
Thus each vertex $y\in W$ of degree at least 2 contributes at
least 1 to sum \eqref{sd}. This yields
%
\begin{prp}\label{d2-4}
  Let $G$ be a bipartite graph on $v+w$ vertices that contains no
  cycle of length 4. 
  \begin{itemize}
  \item[$(i)$] If $w>\binom v2$, then there are at least
  $w-\binom v2$ vertices in $W$ of degree 0 or 1.
  \item[$(ii)$] If its minimal degree is at least 2, then
  $w\le\binom v2$ and $v\le\binom w2$.
\end{itemize}
\end{prp}
If $G$ contains no cycle of length 4 nor 6, then $G'$ contains
no triangle and its size is at most $\lfloor v^2/4\rfloor$.
This argument proves
\begin{prp}\label{d2-6}
  Let $G$ be a bipartite graph on $v+w$ vertices that contains no
  cycle of length 4 or 6. 
  \begin{itemize}
  \item[$(i)$]If $w>\lfloor v^2/4\rfloor $, then there are at least
  $\lceil w-v^2/4\rceil$ vertices in $W$ with degree 0 or 1.
  \item[$(ii)$]If its minimal degree is at least 2, then
  $w\le\lfloor v^2/4\rfloor$ and $v\le\lfloor w^2/4\rfloor$.
  \end{itemize}
\end{prp}


















\section{Bipartite graphs of girth six}
\label{sec:4}









\noindent The following estimate is well known as Reiman's inequality, but its cases of
equality were not written down explicitly. Reading the proof of
\cite[Th.~VI.2.6]{bo78}, one gets with \cite[Def.~I.3.1]{bjl99}
%
\begin{prp}\label{prp6}
Let $v\le w$. A graph of girth at least 6 on $v+w$ vertices with $e$
edges satisfies
$$O(v,w,e)=e^2-we-vw(v-1)\le0$$
$$e\le\sqrt{vw(v-1)+w^2/4}+w/2.$$
We have equality if and only if it is the incidence graph of a
Steiner system $S(2,k;v)$ on $v$ points with block degree $k$ given by $wk(k-1)=v(v-1)$.
\end{prp}
%
Note that by symmetry, we also get $O(w,v,e)\le0$, but this is
superfluous by 
%
\begin{lem}
  Let $v\le w$. Let $e$ be the positive root of $X^2-vX-vw(w-1)$. Then
  $O(v,w,e)\ge0$.
\end{lem}
%
\begin{proof}
As $(vw)^2-vvw-vw(w-1)=vw(vw-v-w+1)\ge0$, we have
  $e\le vw$. Therefore
%
\begin{align*}
e^2-we-vw(v-1)&=e^2-ve-vw(w-1)+(v-w)e+vw(w-v)\\&=(vw-e)(w-v)\ge0~.\qedhere
\end{align*}
%
\end{proof}

\begin{rem}
  The case of equality in Prop.~\ref{prp6} may be described further as follows.
  By \cite[Cor.~I.2.11]{bjl99}, every vertex in $V$ has same degree
  $r$ and every vertex in $W$ has same degree $k$ with
\begin{equation}
  \label{eq:bl}
  k-1\mid v-1~\text{and}~k(k-1)\mid v(v-1), 
\end{equation}
so that $v=1+r(k-1)$ and $k\mid r(r-1)$. For given $k$, this set of
conditions is in fact sufficient for the existence of an extremal
graph for large $r$: this is Wilson's Theorem
\cite[Th.~XI.3.8]{bjl99}. For example, we have the following complete
sets of parameters $(v,w,r,k)$:
$$
(1+r(k-1),r(1+r(k-1))/k,r,k)~\text{for}~1\le k\le 5~\text{and}~k\mid r(r-1).$$
The first set of parameters satisfying
\eqref{eq:bl} for which an extremal graph does not exist is
$(36,\allowbreak42,\allowbreak7,\allowbreak6)$. Consult \cite[Table A1.1]{bjl99} for all known block
designs with $r\le 17$. \cite[Table A5.1]{bjl99} provides the
following sets of parameters $(v,w,r,k)$ for block designs: given any
prime power $q$ and natural number $n$, given $t\le s$,
$$\Bigl(q^n,q^{n-1}\frac{q^n-1}{q-1},\frac{q^n-1}{q-1},q\Bigr)\ ,\ 
\Bigl(\frac{q^{n+1}-1}{q-1},\frac{q^{n+1}-1}{q^2-1}\frac{q^n-1}{q-1},\frac{q^n-1}{q-1},q+1\Bigr)\ ,$$
$$\bigl(q^3+1,q^2(q^2-q+1),q^2,q+1\bigr)\ ,\ 
\bigl(2^{t+s}-2^s+2^t,(2^s+1)(2^s-2^{s-t}+1),2^s+1,2^t\bigr)\ .$$
\end{rem}

The following proposition provides a simpler but coarser bound.
%
\begin{prp}
%
  Let $G$ be a bipartite graph on vertex classes $V$ and $W$ with
  $\#V=v$ and $\#W=w$ without cycles of length 4. Its size satisfies
$$e\le\begin{cases}
\sqrt{2vw(v-1)}&\text{if $w\le v(v-1)/2$}\\
v(v-1)/2+w&\text{otherwise.}
\end{cases}
$$
We have optimality in the second alternative for the
bipartite expansion of a complete graph on $V$ as described in Section
\ref{sec:c1}, on which we add
$w-v(v-1)/2$ new edges by connecting any vertex of $V$ to $w-v(v-1)/2$ new vertices in colour class $W$.
%
\end{prp}
%
\begin{proof}
  By Proposition \ref{d2-4}, if $w>v(v-1)/2$, then
  $w-v(v-1)/2$ vertices in $W$ have degree 0 or 1. If we remove
  them, we remove at most $w-v(v-1)/2$ edges and the remaining graph
  has at most $v(v-1)$ edges because $O(v,v(v-1)/2,v(v-1))=0$. The
  first alternative follows from
\begin{equation*}
O(v,w,\sqrt{2vw(v-1)})=w\sqrt{v(v-1)}(\sqrt{v(v-1)}-\sqrt{2w}).\qedhere
\end{equation*}
\end{proof}












\section{Bipartite graphs of girth eight}
\label{sec:6}













\subsection{Statement of the theorem}
\label{sec:ss1}














\noindent Consult \cite[Def.~1.3,1]{va98} for the definition of  generalised
polygons.
%
\begin{thm}\label{thm:1}
  Let $G$ be a bipartite graph on vertex classes $V$ and $W$ with
  $\#V=v$ and $\#W=w$. If $G$ contains no cycle of length 4 or 6,
  then its size $e$ satisfies
  \begin{equation}
    \label{bpi}
    P(v,w,e)=e^3-(v+w)e^2+2vwe-v^2w^2\le 0.
  \end{equation}
  We have equality exactly in two cases:
  \begin{itemize}
  \item [$(i)$]if $G$ is the complete bipartite graph and $v=1$ or $w=1$;
  \item [$(ii)$]if $G$ is the incidence graph of a  generalised quadrangle.
 \end{itemize}
\end{thm}

\begin{rem}
  Let us first note that this polynomial has exactly one positive
  root in $e$ for positive $v,w$. It suffices to this purpose to show
  that its discriminant is negative. This is $-v^2w^2D$ with
  $$D=27p^2+4s^3-36sp-4s^2+32p,~ s=v+w,\,p=vw.$$
  Let us study
  this quantity for $s\ge2$, $p\ge s-1$. We have
  $$
  \frac{dD}{dp}=54p-36s+32\ge54p-36(p+1)+32=18p-4>0,
  $$
  so that its minimum satisfies $p=s-1$, which implies
  $D=(4s-5)(s-1)^2\ge3$. Therefore Inequality \eqref{bpi} is
  equivalent to an inequality of form $e\le e(v,w)$.
\end{rem}

\begin{rem}
  The case of equality in Th.~\ref{thm:1} implies the following:
  every vertex in $V$ has same degree $s+1$ and every vertex in $W$
  has same degree $t+1$. By \cite[Cor.~1.5.5, Th.~1.7.1]{va98},
  $s+t\mid st(1+st)$ and
  $$v=(t+1)(1+st)~,~w=(s+1)(1+st)~,~e=(s+1)(t+1)(1+st).$$
  Let us
  suppose, by symmetry, that $s\le t$. If $s=0$, we get case $(i)$. If
  $s=1$, we obtain exactly the examples of extremal graphs produced by
  de Caen and Sz{\'e}kely: $W$ consists of $t+1$ horizontal lines and
  as much vertical lines and $V$ is the set of $(t+1)^2$ intersection
  points and $G$ is the point-line incidence graph of this grid (this
  is also the bipartite expansion of a complete bipartite graph on
  $(t+1)+(t+1)$ vertices.)  Otherwise $s,t\ge2$ and $G$ is in fact the
  incidence graph of a generalised quadrangle, so that by
  \cite[Th.~1.7.2]{va98}, $t\le s^2$. Let $q$ be a prime power. Then
  there are generalised quadrangles with set of parameters $(s,t)$ any
  of $(q,q)$, $(q,q^2)$, $(q^2,q^3)$, $(q-1,q+1)$; all known ones fit
  in this list. In particular, by \cite[Th.~1.7.9]{va98}, if $t\ge
  s=2$, then $t=2$ or $t=4$ and in each case there is exactly one
  extremal graph. By \cite[Sec.~1.7.11]{va98}, if $t\ge s=3$, then
  there is a (unique) extremal graph exactly if $t=3,5,9$. There is a
  unique extremal graph with $s=t=4$. It is open whether there exists
  a generalised quadrangle with $s=4$ and $t\in\{11,12\}$.
\end{rem}

\subsection{A generalisation of an inequality of Atkinson et al.}
\label{sec:ss2}











\noindent We first need an optimal lower bound on the number of paths of length
3.  Let us prove the following inequality.

\begin{thm}\label{AWM}
  Let $(a_{ij})_{1\le i\le v,1\le j\le w}$ be a matrix of nonnegative
  coefficients and $\rho,\gamma\ge0$. Let 
\begin{equation}
  \label{eq:part}
  a_{i\star }=\sum_{j=1}^wa_{ij},~ 
  a_{\star j}=\sum_{i=1}^va_{ij}, ~ e=\sum_{i=1}^n\sum_{j=1}^va_{ij}.
\end{equation}
  If $a_{i\star }\ge2\rho$ and
  $a_{\star j}\ge2\gamma$, then
  \begin{equation}
    \label{eq:awmbis}
    \phi=\sum_{i=1}^v\sum_{j=1}^wa_{ij}
  (a_{i\star}-\rho)(a_{\star j}-\gamma)\ge 
e(e/v-\rho)(e/w-\gamma),
  \end{equation}
  equality holding exactly if $a_{i\star }$ and $a_{\star j}$ are constant.
\end{thm}
%
This refines the inequality in \cite{awm60}, which states
\begin{equation}
  \label{eq:awm}
  \psi=\sum_{i=1}^v\sum_{j=1}^wa_{ij}a_{i\star}a_{\star j}\ge {e^3/ vw}
\end{equation}
as, by the Arithmetic-Quadratic Mean Inequality,
\begin{equation}
  \label{eq:refin}
  \phi-\psi=-\gamma\sum_{i=1}^va_{i\star}^2-\rho\sum_{j=1}^wa_{\star
  j}^2+\rho\gamma e\le e(-\gamma e/v-\rho e/w+\rho\gamma)
\end{equation}


\begin{rem}
  If $v=w$ and $a$ is diagonal, Inequality \eqref{eq:awm} is the
  Arithmetic-Cubic Mean Inequality and Inequality \eqref{eq:awmbis}
  becomes
$$
\frac 1v\sum_{i=1}^va_{ii}(a_{ii}-\rho)(a_{ii}-\gamma)\ge\frac
ev\frac{e-v\rho}v\frac{e-v\gamma}v,$$ which is true by Chebyshev's
Inequality \cite[Th.~43]{hlp52} if $a_{ii}\ge\rho$ and
$a_{ii}\ge\gamma$. For our ``non commutative Chebyshev Inequality'',
the conditions $a_{i\star }\ge2\rho$ and $a_{\star j}\ge2\gamma$
cannot be weakened to $a_{i\star }\ge\rho$ and $a_{\star j}\ge\gamma$,
as we have the following counterexamples:
$$\begin{pmatrix}2&5\\4&0\end{pmatrix}
,\begin{pmatrix}0&1&1\\1&0&0\\1&0&0\end{pmatrix}~.$$
\end{rem}

\begin{proof}
  If \eqref{eq:awmbis} is an equality, then so are
  \eqref{eq:refin} and \eqref{eq:awm} and our case of equality
  follows from the identical case of equality in \cite{awm60}, whose
  proof we now imitate. We shall suppose that $a_{i\star }>2\rho$ or
  $a_{\star j}>2\gamma$, so that the whole inequality follows by
  continuity. Fix $e$ and suppose that under this condition the
  $a_{ij}$ are chosen so to minimise $\phi$. We may suppose that the
  rows and the columns have been permuted such that the sequences
  $(a_{i\star})$ and $(a_{\star j})$ are nondecreasing:
\begin{equation}\label{dec}
a_{1\star}\le\dots\le a_{v\star},~ 
a_{\star 1}\le\dots\le a_{\star w}.
\end{equation}
If one of these sequences is constant, the inequality follows by the
Arithmetic-Quadratic inequality (and the case of equality is easy).
Let us suppose that this is not so.

One can suppose that $a_{1w}$ and $a_{v1}$ are positive.  Let us show
the argument for $a_{1w}$. If $a_{1w}=0$, there are $k,l$ such that
$a_{1k},a_{lw}>0$. Make a perturbation by adding $\alpha$ to $a_{1w}$
and to $a_{lk}$ and subtracting $\alpha$ to $a_{1k}$ and to $a_{lw}$.
The row and column sums $a_{i\star }$ and $a_{\star j}$ are unaltered
and $\phi$ increases of
\begin{eqnarray*}
\Delta\phi&=&\alpha\bigl(
   (a_{1\star}-\rho)(a_{\star w}-\gamma)+(a_{l\star}-\rho)(a_{\star k}-\gamma)\\
&&\mathop{-}(a_{1\star}-\rho)(a_{\star k}-\gamma)-(a_{l\star}-\rho)(a_{\star m}-\gamma)\bigr)\\
&=&\alpha
(a_{1\star}a_{\star w}+a_{l\star}a_{\star k}-a_{1\star }a_{\star k}-a_{l\star }a_{\star w})\\
&=&\alpha(a_{1\star}-a_{l\star})(a_{\star w}-a_{\star k}),
\end{eqnarray*}
so that $\phi$ does not increase.

Now make the following perturbation: add $2\alpha$ to $a_{11}$ and
subtract $\alpha$ to $a_{1w}$ and to $a_{v1}$. Let us compute the
differential of $\phi$: as
$$\frac{d\phi}{da_{rc}} = (a_{r\star }-\rho)(a_{\star c}-\gamma) +
\sum_{i=1}^va_{ic}(a_{i\star }-\rho) + \sum_{j=1}^wa_{rj}(a_{\star
  j}-\gamma),$$
\begin{eqnarray*}
d\phi&=&d\alpha\Bigl(2\frac{d\phi}{da_{11}}-\frac{d\phi}{da_{1w}}-\frac{d\phi}{da_{v1}}\Bigr)\\
&=&d\alpha\Bigl((a_{1\star }-\rho)(a_{\star 1}-a_{\star w}) +
(a_{1\star }-a_{v\star })(a_{\star 1}-\gamma) +
\sum_{i=1}^va_{i1}(a_{i\star }-\rho) 
\\&&\qquad\qquad\mathop{+}\sum_{j=1}^wa_{1j}(a_{\star j}-\gamma) - 
\sum_{i=1}^va_{iw}(a_{i\star }-\rho) - 
\sum_{j=1}^wa_{vj}(a_{\star j}-\gamma)\Bigr)\\
\end{eqnarray*}
For positive $d\alpha$, we have by \eqref{dec}
\begin{eqnarray*}
d\phi&\le&d\alpha\bigl(
(a_{1\star }-\rho)(a_{\star 1}-a_{\star w}) +
(a_{1\star }-a_{v\star })(a_{\star 1}-\gamma) +
a_{\star 1}(a_{v\star }-\rho)\\
&&\qquad\qquad\qquad\quad
\mathop{+}a_{1\star }(a_{\star w}-\gamma)-a_{\star w}(a_{1\star }-\rho)
-a_{v\star }(a_{\star 1}-\gamma)\bigr)\\
&=&d\alpha\bigl(
(a_{1\star }-2\rho)(a_{\star 1}-a_{\star w})+(a_{\star
  1}-2\gamma)(a_{1\star }-a_{v\star })\bigr)\\
&<&0\,,
\end{eqnarray*}
which contradicts the minimum hypothesis.
\end{proof}

\begin{cor}\label{path3}
  Let $G$ be a bipartite graph on $v+w$ vertices and of minimal degree
  2. Then the number of paths of length 3 in $G$ is at least
  $e(e/v-1)(e/w-1)$. This bound is achieved exactly if the graph is
  regular for each of its two colours.
\end{cor}
%
\begin{proof}
  A path of length 3 is a sequence of 4 vertices
  $(x,y,z,t)$ with no repetition such that
  $$\{x,y\},\{y,z\},\{z,t\}\in G.$$
  Given two adjacent vertices $y$
  and $z$, the number of paths $(x,y,z,t)$ makes $(d(y)-1)(d(z)-1)$,
  where $d$ denotes the degree of a vertex. Therefore the number of
  all paths of length 3 is $$\sum_{\{y,z\}\in G}(d(y)-1)(d(z)-1).$$
  Let $(a_{ij})_{1\le i\le v,1\le j\le w}$ be the reduced incidence
  matrix of $G$: $a_{ij}=1$ if the $i$th vertex of the first class is
  adjacent to the $j$th vertex of the second class; otherwise
  $a_{ij}=0$. Then this sum is
\begin{equation}
  \label{eq:sum}
  \sum_{i=1}^v\sum_{j=1}^wa_{ij}(a_{i\star }-1)(a_{\star j}-1),
\end{equation}
so that it suffices to take $\rho=\gamma=1$ in Th.~\ref{AWM}.
\end{proof}






\subsection{Proof of Theorem \ref{thm:1}}
\label{sec:proof}

\noindent The case of equality follows from \cite[Lemma~1.4.1]{va98} because its
axiom $(i)$ is exactly what makes Bound \eqref{eq:bound} an
equality.

I now follow the proof of \cite[Th.~1]{ds97}. If $G$ contains no cycle
of length 4, there is no path of length 3 between two adjacent
vertices; if $G$ contain no cycle of length 6, there is at most one
path of length 3 between non-adjacent vertices of different colour.
Therefore the sum \eqref{eq:sum} is bounded by
\begin{equation}
  \label{eq:bound}
  vw-e~\text{with}~e=\sum_{i=1}^n\sum_{j=1}^va_{ij}.
\end{equation}
By Corollary~\ref{path3}, if all the vertices of $G$ have degree at
least two, one has
$$vw-e\ge e^3/vw-(1/v+1/w)e^2+e$$
and therefore \eqref{bpi}.  In
order to get rid of this degree condition, we have to do an induction
on the sum $s=v+w$ of the number of columns and the number of rows of
the incidence matrix. If $v=1$, then $P(v,w,e)=(e-w)(e^2-e+w)$, so
that the inequality states $e\le w$, which is trivial; symmetrically
for $w=1$. Suppose the result is true for all $v\times w$ incidence
matrices with $v+w=s$.  Consider now a $v\times w$ incidence matrix
with $v+w=s+1$ and $v,w\ge2$. If each vertex has degree at least two,
the result is true; otherwise there is a column or a row containing
only zeroes or exactly one ``1''. Apply the induction hypothesis on
the matrix without this row or column: we get $P(v-1,w,e-1)\le0$ or
$P(v,w-1,e-1)\le0$ and we may apply the following growth lemma to
conclude.




\begin{lem}
Let $v,w\ge1$. If $P(v,w,e)\le0$, then $P(v+1,w,e+1)\le0$.
\end{lem}
\begin{proof}
  In fact, one has
  $$P(v+1,w,e+1)-P(v,w,e)=2{e}^{2}+(1-2v)e+(w-{w}^{2})(2v+1)-v,$$
  which is negative as long as
  $$0\le e\le e_0=\bigr(2v-1+\sqrt{ \left (2v+1\right )\left
      (2v+8{w}^{2}-8w+1\right )}\bigl)/4=(2v-1+\Delta)/4.$$
  Let us use
  that $P(v,w,e)$ has a unique root in $e$ and compute $P(v,w,e_0)$.
  This makes

  $$(4vw^2+2w^2+1)\Delta/16+(-16vw^3-8v^2w^2-8w^3+8vw^2+2w^2-2v+4w-1)/16.$$
  Then either the second term in this sum is positive and $P(v,w,e_0)$
  is positive, or the conjugate expression of this sum is positive,
  and the product of the sum with this conjugate expression is
  $$(w-1)^2w^2(8v^3w^2+4v^2w^2-2vw^2+2v^2-w^2-4vw+2v-2w)/8,$$
  which is
  positive if $v,w\ge1$.
\end{proof}


















\subsection{Further remarks}
\label{sec:ss3}












\noindent \begin{rem}
  Theorem \ref{thm:1} does not always give the right order of
  magnitude for the maximal size of a graph of girth 8: as

  $$P(v,w,(vw)^{2/3})=2(vw)^{5/3}-(vw)^{4/3}(v+w))\le2(vw)^{5/3}-2(vw)^{4/3+1/2}\le0,$$
  we expect to find maximal graphs of size $(vw)^{2/3}$: De~Caen and
  Sz\'ekely \cite[Th.~4]{ds97} find a counterexample to this
  expectation if $v$ ``lies in an interval just slightly below''
  $w^2$. They conjecture \cite{ds92} that this is the case as soon as
  $v\gg w^{5/4}$ and $v\ll w^2$.
\end{rem}

In the case of $v=w$, let us give the following approximation for the
real root of the polynomial. For
$$e=v^{4/3}+\frac23v-\frac29v^{2/3}-\frac{20}{81}v^{1/3},$$
$$P(v,v,e)={\frac {40}{243}}{v}^{7/3}+{\frac
  {376}{2187}}{v}^{2}-{\frac {80}{2187}}{v}^{5/3}-{\frac
  {800}{19683}}{v}^{4/3}-{\frac {8000}{531441}}v\ge{\frac
  {129808}{531441}},$$
\begin{eqnarray*}
P(v,v,e-16/81)&=&-\frac8{531441}(v^{1/3}-1)\bigl(
39366{v}^{7/3}+28431{v}^{2}+8262{v}^{5/3}\\
&&\mathop{-}8748{v}^{
4/3}-11880{v}-6560{v}^{2/3}-2432v^{1/3}-512\bigr)\\&\le&0.
\end{eqnarray*}
In particular,
\begin{cor}
  Let $G$ be a bipartite graph of size $e$ with $v$ vertices in each vertex
  class. If the girth of $G$ is at least 8, then
$$e<v^{4/3}+\frac23v-\frac29v^{2/3}-\frac{20}{81}v^{1/3}.$$
\end{cor}



Let us now show that we generalise the following estimations for the
size of bipartite graphs of girth 8 in \cite[Th.~1]{ds97}:
\begin{itemize}
\item[$(i)$]\emph{if the minimal degree of $G$ is at least 2, then
  $e\le2^{1/3}(vw)^{2/3}$;}
\item[$(ii)$]\emph{if $v\preccurlyeq w^2$ or $w\preccurlyeq v^2$, then
$e\preccurlyeq(vw)^{2/3}$.}
\end{itemize}
In fact,
$$P(v,w,2^{1/3}(vw)^{2/3})=(vw)^{4/3}(w^{2/3}-2^{2/3}v^{1/3})(v^{2/3}-2^{2/3}w^{1/3}),$$
which is nonnegative exactly if $v\le w^2/4$ and $w\le v^2/4$ or if
$(v,w)$ is among $\bigl\{(1,1),\allowbreak(1,2),\allowbreak(2,1),\allowbreak(2,2),\allowbreak(3,3)\bigr\}$, and
this is the case by Prop.~\ref{d2-6} if the minimal degree is at least
2.

Furthermore, by Prop.~\ref{d2-6}, if $w>\lfloor v^2/4\rfloor$, then
$\lceil w-v^2/4\rceil$ vertices in
$W$ have degree 0 or 1. If we remove them, we remove at most
$\lceil w-v^2/4\rceil$ edges and the remaining graph has at most
$\lfloor v^2/2\rfloor$ edges because $P(v,\lfloor v^2/4\rfloor,\lfloor v^2/2\rfloor+1)>0$. 
This yields
\begin{prp}
  Let $G$ be a bipartite graph on vertex classes $V$ and $W$ with
  $\#V=v$ and $\#W=w$ without cycles of length 4 and 6. Its size satisfies
$$e\le\begin{cases}
2^{1/3}(vw)^{2/3}&\text{if $\max(v,w)\le \lfloor
  \min(v,w)^2/4\rfloor$}\\ 
\lfloor\min(v,w)^2/4\rfloor+\max(v,w)&\text{otherwise.}
\end{cases}
$$
We have optimality in the second alternative: make
a bipartition $V=V_1\cup V_2$ with $V_1=\lceil v/2\rceil$ and
$V_2=\lfloor v/2\rfloor$, let $G'$ be the complete bipartite graph
on the colour classes $V_1$ and $V_2$, which has $\lfloor
v^2/4\rfloor$ edges. Now consider the bipartite expansion of $G'$,
add $\lceil w-v^2/4\rceil$ new vertices to colour class $W$,
and connect each of them to some vertex of $V$.
\end{prp}
Note that this estimate yields another proof of \cite[Th.~1]{gy97} by
means of \cite[Th.~3]{gy97}.  



\begin{rem}
  Our inequality condenses the following facts about the
  behaviour of $e$ for fixed $w$ and large $v$. If $w\le 3$, then
  extremal graphs of girth 8 do not contain any cycle at all, so
  that their size is $e=v+w-1$; if $v\ge w=4$ and if $v=w=5$, then
  extremal graphs of girth 8 contain exactly one cycle, so that
  their size is $e=v+w$; if $v>w=5$, then extremal graphs of girth 8
  contain exactly one ``$\theta$-graph'', so that their size is
  $e=v+w+1$.
\end{rem}

 









% \def\cprime{$'$} \def\cprime{$'$}
% \begin{thebibliography}{1}

% \bibitem{awm60}
% F.~V. Atkinson, G.~A. Watterson and P.~A.~P. Moran, {\em A matrix inequality},
%   Quart. J. Math. Oxford Ser. (2) 11 (1960),  137--140.

% \bibitem{bjl99}
% T.~Beth, D.~Jungnickel and H.~Lenz, {\em Design theory. Two volumes}, Cambridge
%   University Press, second~ed., 1999.

% \bibitem{bo78}
% B.~Bollob{\'a}s, {\em Extremal graph theory}, Academic Press, 1978.

% \bibitem{ds92}
% D.~de~Caen and L.~A. Sz{\'e}kely, {\em The maximum size of 4- and 6-cycle
%   free bipartite graphs on $m,n$ vertices}, in: Sets, graphs and numbers
%   (Budapest, 1991), North-Holland, 1992,  135--142.

% \bibitem{ds97}
% \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em On dense bipartite
%   graphs of girth eight and upper bounds for certain configurations in planar
%   point-line systems}, J. Combin. Theory Ser. A 77 (1997),  268--278.

% \bibitem{gy97}
% E.~Gy{\"o}ri, {\em ${C}\sb 6$-free bipartite graphs and product representation
%   of squares}, Discrete Math. 165/166 (1997),  371--375.

% \bibitem{hlp52}
% G.~H. Hardy, J.~E. Littlewood and G.~P{\'o}lya, {\em Inequalities}, Cambridge
%   Univ. Press, second~ed., 1952.

% \bibitem{va98}
% H.~van Maldeghem, {\em Generalized polygons}, Birkh\"auser Verlag, 1998.

% \end{thebibliography}



 


%%% Local Variables: 
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\clearpage

%\input worknote 

\chapter{\protect{\texorpdfstring{Ordering simultaneously the columns and lines of
  0{,}1~matrices}{Ordering simultaneously the columns and lines of
  0,1 matrices}}}
\begin{flushright}
  \large with Nikolai Kosmatov.
\end{flushright}\bigskip

\noindent Let $M$ be a $0,1$ matrix with $m$ rows and $n$ columns:
$$
M=(a_{ij})=
\begin{pmatrix}
a_{11} & a_{12} & \ldots & a_{1n}\\
a_{21} & a_{22} & \ldots & a_{2n}\\
\vdots & \vdots &  & \vdots\\
a_{m1} & a_{m2} & \ldots & a_{mn}\\
\end{pmatrix}~\text{with}~a_{ij}\in \{0,\,1\}.
$$

There are $2^{m+n}$ such matrices. We would like to study these
matrices up to a permutation of the rows and a permutation of the
columns, and therefore propose an order on the set of rows and on the
set of columns.

\begin{dfnk}
  Let $L_i=L_i(M)=(a_{i1},\,a_{i2},\,\dots,\,a_{in})$, $1 \leqslant i
  \leqslant m$, be the lines of $M$ and let
  $C_j=C_j(M)=(a_{1j},\,a_{2j},\,\dots,\,a_{mj})$, $1 \leqslant j
  \leqslant n$, be its columns.  The rows (vs.~columns) of $M$
  are \emph{ordered} if they form an increasing sequence with respect
  to the lexicographic order.
\end{dfnk}

Note that if the rows and columns are considered as digit sequences of
binary numbers, the lexicographic order is just the natural order on
${\mathbb N}$.

\begin{thmk}\label{t1}
There is a permutation of rows and columns that orders them
simultaneously. 
% The rows and columns of $M$ can be ordered simultaneously by permuting
% them.
\end{thmk}

\begin{dfnk}
  The numbers $r\,,s \geqslant 1$, $0 = i_0 < i_1 < i_2 < \dots < i_r
  = m$, $0 = j_0 < j_1 < j_2 < \dots < j_s = n$ define a \emph{grid} on
  $m\times n$ that divides $M$ in $r \times s$ block matrices
$$
B_{t,u}=B_{t,u}(M)=
\begin{pmatrix}
  a_{i_{t-1}+1,j_{u-1}+1} & \ldots & a_{i_{t-1}+1,j_u} \\
  \vdots & & \vdots \\
  a_{i_t,j_{u-1}+1} & \ldots & a_{i_t,j_u} \\
\end{pmatrix},~1 \leqslant t \leqslant r,~1 \leqslant u \leqslant
s,
$$
so that
\begin{multline*}
  M=
    \begin{pmatrix}
      B_{1,1}&\dots&B_{1,s}\\
      \vdots&&\vdots\\
      B_{r,1}&\dots&B_{r,s}
    \end{pmatrix}
  = \left(
    \arraycolsep3pt
    \begin{array}{ccc|c|ccc}
      a_{i_0+1,j_0+1} & \ldots & a_{i_0+1,j_1} & \ldots & a_{i_0+1,j_{s-1}+1} & \ldots & a_{i_0+1,j_s} \\
      \vdots & & \vdots & & \vdots & & \vdots \\
      a_{i_1,j_0+1} & \ldots & a_{i_1,j_1} & \ldots & a_{i_1,j_{s-1}+1} & \ldots & a_{i_1,j_s} \\
      \hline
      a_{i_1+1,j_0+1} & \ldots & a_{i_1+1,j_1} & \ldots & a_{i_1+1,j_{s-1}+1} & \ldots & a_{i_1+1,j_s} \\
      \vdots & & \vdots & & \vdots & & \vdots \\
      a_{i_2,j_0+1} & \ldots & a_{i_2,j_1} & \ldots & a_{i_2,j_{s-1}+1} & \ldots & a_{i_2,j_s} \\
      \hline
      \vdots & & \vdots &  & \vdots & & \vdots \\
      \hline
      a_{i_{r-1}+1,j_0+1} & \ldots & a_{i_{r-1}+1,j_1} & \ldots & a_{i_{r-1}+1,j_{s-1}+1} & \ldots & a_{i_{r-1}+1,j_s} \\
      \vdots & & \vdots & & \vdots & & \vdots \\
      a_{i_r,j_0+1} & \ldots & a_{i_r,j_1} & \ldots & a_{i_r,j_{s-1}+1} & \ldots & a_{i_r,j_s} \\
    \end{array} 
  \right).
\end{multline*}
A permutation of lines $L_{p_1}$ and $L_{p_2}$ is \emph{admissible}
if $i_{t-1} < p_1 \leqslant i_t$ and $i_{t-1} < p_2 \leqslant i_t$ for
some $t\in\{1,2,\dots,r\}$; otherwise, it is \emph{inadmissible}.  A
permutation of columns $C_{q_1}$ and $C_{q_2}$ is \emph{admissible}
if $j_{u-1} < q_1 \leqslant j_u$ and $j_{u-1} < q_2 \leqslant j_u$ for
some $u\in\{1,2,\dots,s\}$; otherwise, it is \emph{inadmissible}.  In
other words, admissible permutations of rows and columns only permute
matrix coefficients inside the block matrices defined by the grid.
Matrix $M$ is \emph{ordered with respect to the grid} if
\begin{gather*}
  L_{i_0+1} \leqslant L_{i_0+2} \leqslant\dots\leqslant L_{i_1},~
  L_{i_1+1} \leqslant\dots\leqslant L_{i_2},~\dots,~
  L_{i_{r-1}+1} \leqslant\dots\leqslant L_{i_r};\\
  C_{j_0+1} \leqslant C_{j_0+2} \leqslant\dots\leqslant C_{j_1},~
  C_{j_1+1} \leqslant\dots\leqslant C_{j_2},~\dots,~ C_{j_{s-1}+1}
  \leqslant\dots\leqslant C_{j_s},
\end{gather*}
where $\leqslant$ is the lexicographic order.
\end{dfnk}

Theorem~\ref{t1} is an
immediate consequence of Lemma~\ref{l2}: it suffices to consider
the trivial grid given by $r=s=1$, $0=i_0<i_1=m$, $0=j_0<j_1=n$.

\begin{lemk} \label{l2}
The rows and columns of a $0,1$ matrix may be simultaneously ordered
with respect to a given grid by admissible permutations of its rows
and columns. 
\end{lemk}

\begin{proof}[Proof by induction]
  If $m=1$ or $n=1$, this is trivial. Let us
  suppose $m,n>1$ and let us construct the first line and then the
  first column of the reordered matrix. Let
$$
A=
\begin{pmatrix}
  \sum_{q=j_0+1}^{j_1} a_{1q} & \sum_{q=j_1+1}^{j_2} a_{1q} & \ldots & \sum_{q=j_{s-1}+1}^{j_s} a_{1q}\\
  \sum_{q=j_0+1}^{j_1} a_{2q} & \sum_{q=j_1+1}^{j_2} a_{2q} & \ldots & \sum_{q=j_{s-1}+1}^{j_s} a_{2q}\\
  \vdots & \vdots &  & \vdots\\
  \sum_{q=j_0+1}^{j_1} a_{i_1q} & \sum_{q=j_1+1}^{j_2} a_{i_1q} & \ldots & \sum_{q=j_{s-1}+1}^{j_s} a_{i_1q}\\
\end{pmatrix}
$$
be the matrix of numbers of ones in the lines of the block matrices
$B_{1,u}$, $1 \leqslant u \leqslant s$.  Let $L_v(A)$ be the minimal
line of $A$: we permute $L_1(M)$ with $L_v(M)$ and still call $M$ the
resulting matrix. Then
$$
\sum_{q=j_0+1}^{j_1} a_{1q} \leqslant \sum_{q=j_0+1}^{j_1} a_{iq},
$$
for each $i\in\{1,\,2,\,\dots,i_1\}$, so that either some $a_{1j}$,
$1\leqslant j \leqslant j_1$, vanishes, either $B_{1,1}$ is a matrix
of ones. Let us permute the columns of $M$ in such a way that that the
first lines of all the resulting block matrices $B_{1,u}$, $1
\leqslant u \leqslant s$, consist in a block of zeroes followed by a
block of ones:
$$
B_{1,u}=
\begin{pmatrix}
  0 & 0 & \ldots & 0 & 0 & 1 & 1 & \ldots & 1 & 1 \\
  {*} & * & \ldots & * & * & * & * & \ldots & * & * \\
  \vdots & \vdots &  & \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots \\
  {*} & * & \ldots & * & * & * & * & \ldots & * & *
\end{pmatrix}.
$$
We still call $M$ the resulting matrix. We now refine the grid on
$m\times n$ in such a way that the first lines of the block matrices
of $M$ with respect to the new grid either consist only of zeroes or
consist only of ones, as follows:
\begin{equation}\label{form}
  \left(
    \begin{array}{ccc|ccc||c||ccc|ccc}
      0 & \ldots & 0 & 1 & \ldots & 1 & \ldots & 0 & \ldots & 0 & 1 & \ldots & 1  \\
      {*} & \ldots & * & * & \ldots & * & \ldots & * & \ldots & * & * & \ldots & * \\
      \vdots & & \vdots & \vdots & & \vdots & & \vdots & & \vdots & \vdots & & \vdots \\
      {*} & \ldots & * & * & \ldots & * & \ldots & * & \ldots & * & * & \ldots & * \\
      \hline
      \hline
      {*} & \ldots & * & * & \ldots & * & \ldots & * & \ldots & * & * & \ldots & * \\
      \vdots & & \vdots & \vdots & & \vdots & & \vdots & & \vdots & \vdots & & \vdots \\
    \end{array}
  \right),
\end{equation}
where the double lines belong to the original grid and the simple ones
have been added now. Let this new grid be defined by the numbers $r,s'
\geqslant 1$, $0 = i_0 < i_1 < i_2 < \dots < i_r = m$, $0 = j_0' <
j_1' < j_2' < \dots < j_{s'}' = n$, and let $B_{t,u}'$,
$t\in\{1,2,\dots,r\}$, $u\in\{1,2,\dots,s'\}$, be the block matrices
with respect to it.  The block matrix $B_{1,1}'$ has one of the two
following forms:
$$
\begin{pmatrix}
  0 & 0 & \ldots & 0 & 0 \\
  {*} & * & \ldots & * & * \\
  \vdots & \vdots & & \vdots & \vdots \\
  {*} & * & \ldots & * & *
\end{pmatrix}
~\text{or}~
\begin{pmatrix}
  1 & 1 & \ldots & 1 & 1 \\
  1 & 1 & \ldots & 1 & 1 \\
  \vdots & \vdots & & \vdots & \vdots \\
  1 & 1 & \ldots & 1 & 1
\end{pmatrix}.
$$

We now proceed similarly with the first column. Let
$$
B=
\begin{pmatrix}
  \sum_{p=i_0+1}^{i_1} a_{p1} & \sum_{p=i_0+1}^{i_1} a_{p2} & \ldots & \sum_{p=i_0+1}^{i_1} a_{pj_1'}\\
  \sum_{p=i_1+1}^{i_2} a_{p1} & \sum_{p=i_1+1}^{i_2} a_{p2} & \ldots & \sum_{p=i_1+1}^{i_2} a_{pj_1'}\\
  \vdots & \vdots &  & \vdots\\
  \sum_{p=i_{r-1}+1}^{i_r} a_{p1} & \sum_{p=i_{r-1}+1}^{i_r} a_{p2} &
  \ldots & \sum_{p=i_{r-1}+1}^{i_r} a_{pj_1'}
\end{pmatrix}
$$
be the matrix of numbers of ones in the columns of the block matrices
$B'_{t,1}$, $1 \leqslant t \leqslant r$.  Let $C_v(B)$ be the minimal
column of $B$: we permute $C_1(M)$ with $C_v(M)$ and still call $M$
the resulting matrix. Let us permute the rows of $M$ in such a way
that the first columns of all the block matrices $B'_{t,1}$, $1
\leqslant t \leqslant r$, consist in a block of zeroes followed by a
block of ones:
$$
B_{t,1}'(M)=
\begin{pmatrix}
  0 & * & \ldots & * \\
  \vdots & \vdots & & \vdots \\
  0 & * & \ldots & * \\
  1 & * & \ldots & * \\
  \vdots & \vdots & & \vdots \\
  1 & * & \ldots & *
\end{pmatrix}
$$
The form of $B_{1,1}'$ shows that this can be done without permuting
the first row, so we do not.  We still call the resulting matrix $M$.
We now refine the grid on $m\times n$ in such a way that the first
columns of the block matrices with respect to the new grid either
consist only of zeroes or consist only of ones, so that also the
tranpose of $M$ has a form as in \eqref{form}.
% $$
% B'_{t,1}(M)=\left(
%   \begin{array}{c}
%   B''_{t,1}(M)\\
%   % \hline
%   % B''_{t,2}(M)\\
% \end{array}
% \right)=\left(
%   \begin{array}{cccc}
% 0 & * & \ldots & * \\
% \vdots & \vdots & & \vdots \\
% 0 & * & \ldots & * \\
% \hline
% 1 & * & \ldots & * \\
% \vdots & \vdots & & \vdots \\
% 1 & * & \ldots & * \\
% \end{array}
% \right)
% $$
Let the new grid be defined by the numbers $r',s' \geqslant 1$, $0 =
i_0' < i_1' < i_2' < \dots < i_{r'}' = m$, $0 = j_0' < j_1' < j_2' <
\dots < j_{s'}' = n$.% , and let $B_{t,u}''(M)$,
% $t\in\{1,2,\dots,r'\}$, $u\in\{1,2,\dots,s'\}$, be the block
% matrices with respect to it.

Let $N$ be the $m-1\times n-1$ submatrix in the lower right corner of
$M$. By induction hypothesis, $N$ may be ordered by admissible
permutations with respect to the grid on $m-1\times n-1$ induced by
the grid just defined. These permutations are also admissible with
respect to the latter grid and do not permute the first line and the
first column of $M$. Let us again call the resulting matrices $M$ and
$N$.

As the rows and columns of $N$ are ordered, we have for $M$
\begin{gather*}
  L_{i_0+2} \leqslant\dots\leqslant L_{i_1},~ L_{i_1+1}
  \leqslant\dots\leqslant L_{i_2},~\dots,~
  L_{i_{r-1}+1} \leqslant\dots\leqslant L_{i_r};\\
  C_{j_0+2} \leqslant\dots\leqslant C_{j_1},~ C_{j_1+1}
  \leqslant\dots\leqslant C_{j_2},,~\dots,~ C_{j_{s-1}+1}
  \leqslant\dots\leqslant C_{j_s},
\end{gather*}
and our choice for the first line and the first column of $M$ ensures
that
\begin{equation*}
  L_{i_0+1} \leqslant L_{i_0+2},~  C_{j_0+1} \leqslant C_{j_0+2}.\qedhere
\end{equation*}
\end{proof}


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\clearpage

\chapter[Transfer of Fourier  into Schur multipliers and sumsets in a group]{Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group}\label{sumsets}

\begin{flushright}
  \large with \'Eric Ricard
\end{flushright}\smallskip

\begin{abstract}
\noindent We inspect the relationship between relative Fourier multipliers on noncommutative Le\-besgue-Orlicz spaces of a discrete group $\Ga$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets; unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\La\subseteq\Ga$; the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2; the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.
\end{abstract}

\section{Introduction}

\noindent Let $\La$ be a subset of $\Z$ and let $x$ be a bounded measurable
function on the circle $\T$ with Fourier spectrum in $\La$:
we write $x\in\Ell^\infty_\Lambda$, $x\sim\sum_{k\in\La}x_kz^k$. The matrix of the associated operator
$y\mapsto xy$ on $\Ell^2$ with respect to its trigonometric basis is the Toeplitz matrix
\begin{equation*}
  (x_{r-c})_{(r,c)\in\Z\times\Z}=
  \bordermatrix{
    &\scriptstyle\cdots&\scriptstyle1&\scriptstyle0&\scriptstyle-1&\scriptstyle\cdots\cr
    \hfil\scriptstyle\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\cr
    \hfil\scriptstyle1&\ddots&x_0&x_{1}&x_{2}&\ddots\cr
    \hfil\scriptstyle0&\ddots&x_{-1}&x_0&x_{1}&\ddots\cr
    \hfil\scriptstyle-1&\ddots&x_{-2}&x_{-1}&x_0&\ddots\cr
    \hfil\scriptstyle\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\cr}
\end{equation*}
with support in $\HLa=\{(r,c):r-c\in\La\}$. 

This is an entry point to the interplay between harmonic analysis and
operator theory. In the general case of a discrete group $\Ga$, the
counterpart to a bounded measurable function is defined as a bounded
operator on $\ell^2_\Gamma$ whose matrix has the form
$(x_{rc^{-1}})_{(r,c)\in\Ga\times\Ga}$ for some sequence
$(x_\ga)_{\ga\in\Ga}$. This will be the framework of the body of this
article, while the introduction sticks to the case $\Ga=\Z$.

We are concerned with two kinds of multipliers. A sequence
$\ph=(\ph_k)_{k\in\La}$ defines
\begin{itemize}
\item the relative Fourier multiplication operator on trigonometric
  polynomials with spectrum in $\La$ by
  \begin{equation}
    \sum_{k\in\La}x_kz^k\mapsto\sum_{k\in\La}\ph_kx_kz^k;\label{rfmo}
  \end{equation}
\item the relative Schur multiplication operator on finite matrices
  with support in $\HLa$ by
  \begin{equation}
    (x_{r,c})_{(r,c)\in\Z\times\Z}\mapsto(\Hph_{r,c}x_{r,c})_{(r,c)\in\Z\times\Z}\label{rsmo}
  \end{equation}
  where $\Hph_{r,c}=\ph_{r-c}$.  
\end{itemize}
Marek Bo\.zejko and Gero Fendler proved that these two multipliers
have the same norm. The operator~\eqref{rfmo} is nothing but the
restriction of~\eqref{rsmo} to Toeplitz matrices. They noted that it
is automatically \emph{completely} bounded: it has the same norm when
acting on trigonometric series with operator coefficients $x_k$, and
this permits to remove this restriction. Schur multiplication is also
automatically completely bounded.

This observation has been extended by Gilles Pisier to multipliers
acting on a translation invariant Lebesgue space $\Ell^p_\La$ and on
the subspace $\Sch^p_{\mkern-3mu\HLa}$ of elements of a
Schatten-von-Neumann class supported by $\HLa$, respectively: it
yields that the complete norm of a relative Schur
multiplier~\eqref{rsmo} remains bounded by the complete norm of the
relative Fourier multiplier~\eqref{rfmo}.

But $\Ell^p_\La$ is not a subspace of $\Sch^p_{\mkern-3mu\HLa}$, so
that a relative Fourier multiplier may not be viewed anymore as the
restriction of a relative Schur multiplier to Toeplitz matrices. We
point out that this difficulty may be overcome by using Szeg\H{o}'s
limit theorem: a bounded measurable real function is the
$\textrm{weak}^*$ limit of the normalised counting measure of
eigenvalues of finite truncates of its Toeplitz matrix. Note that
other types of approximation are also available, as the completely
positive approximation property and Reiter sequences combined with
complex interpolation: they are compared in Section~\ref{sec:loc} in
terms of local embeddings of $\Ell^p$ into $\Sch^p$. They are more
canonical than Szeg\H{o}'s limit theorem, but give no access to
general Orlicz norms.

\begin{thm}
  Let $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing function
  vanishing only at~0. The norm of the following operators is equal:
  \begin{itemize}
  \item the relative Fourier multiplication operator~\eqref{rfmo} on
    the Lebesgue-Orlicz space $\Ell^\psi_\La(\Sch^\psi)$ of
    $\Sch^\psi$-valued trigonometric series with spectrum in $\La$;
  \item the relative Schur multiplication operator~\eqref{rsmo} on the
    Schatten-von-Neu\-mann-Orlicz class $\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)$ of
    $\Sch^\psi$-valued matrices with support in $\HLa$.
  \end{itemize}

\end{thm}

Look at Theorem~\ref{trans2} for the precise statement in the general
case of an amenable group $\Ga$, for which a block matrix variant of
Szeg\H{o}'s limit theorem in the style of Erik B\'edos \cite{be97},
Theorem~\ref{sz}, is available.

An application of this theorem to the class of all unimodular Fourier
multipliers yields a transfer of lacunary subsets into lacunary matrix
patterns. Call $\La$ \emph{unconditional in $\Ell^p$} if
$(z^k)_{k\in\La}$ is an unconditional basis of $\Ell^p_\La$, and
$\HLa$ \emph{unconditional in $\Sch^p$} if the sequence
$(\e_q)_{q\in\HLa}$ of elementary matrices is an unconditional basis
of $\Sch^p_{\mkern-3mu\HLa}$. These properties are also known as
$\Lambda(\max(2,p))$ and $\sigma(p)$, respectively; they have natural
``complete'' counterparts that are also known as $\Lambda(p)_\cb$
($\mathrm{K}(p)_\cb$ if $p\le2$) and $\sigma(p)_\cb$, respectively: see Definitions
\ref{unclp:def}~and~\ref{uncsp:def}.
\begin{cor}
  If $\HLa$ is unconditional in $\Sch^p$, then $\La$ is unconditional
  in $\Ell^p$. \ $\HLa$ is completely unconditional in $\Sch^p$ if and
  only if $\La$ is completely unconditional in $\Ell^p$.
\end{cor}
Look at Proposition~\ref{trans:set} for the precise statement in the
general case of a discrete group $\Ga$.

The two most prominent multipliers are the Riesz projection
and the Hilbert transform. The first consists in letting $\ph$~be the
indicator function of nonnegative integers and transfers into the
upper triangular truncation of matrices. The second corresponds to the
sign function and transfers into the Hilbert matrix transform. We
obtain the following partial results.

\begin{thm}\label{rhtrht}
  The norm of the matrix Riesz projection and of the matrix
  Hilbert transform on $\Sch^\psi(\Sch^\psi)$ coincide with their norm on~$\Sch^\psi$.
  \begin{itemize}
  \item If $p$~is a power of~2, the norm of the matrix Hilbert
    transform on~$\Sch^p$ is $\cot(\pi/2p)$.
  \item The norm of the matrix Riesz projection on~$\Sch^4$ is~$\sqrt2$.
  \end{itemize}
\end{thm}

The transfer technique lends itself naturally to the case where $\La$
contains a sumset $\Row+\Col$: if subsets $\Row'$~and~$\Col'$ are
extracted so that the $r+c$ with $r\in\Row'$ and $c\in\Col'$ are
pairwise distinct, they may play the r\^ole of rows and columns.  Here
are consequences of the conditionality of the sequence of elementary
matrices $\e_{r,c}$ in $\Sch^p$ for $p\ne2$ and of the unboundedness
of the Riesz transform on $\Sch^1$ and $\Sch^\infty$, respectively.

\begin{thm}If $(z^k)_{k\in\La}$ is a completely unconditional basis of
  $\Ell^p_\La$ with $p\ne2$, then $\La$ does not contain sumsets
  $\Row+\Col$ of arbitrarily large sets.
  \begin{itemize}
  \item If $\Ell^1_\La$ admits some completely unconditional
    approximating sequence, or
  \item if the space $\Cont_\La$ of continuous functions with spectrum
    in $\La$ admits some unconditional approximating sequence,
  \end{itemize}    
  then $\La$ does not contain
  the sumset $\Row+\Col$ of two infinite sets.
\end{thm}

The proof of the second part of this theorem consists in constructing
infinite subsets $\Row'$~and~$\Col'$, and skipped block
sums $\sum(T_{k_{j+1}}-T_{k_j})$ of a given approximating sequence that
act like the projection on the ``upper triangular'' part
of~$\Row'+\Col'$.  Look at Proposition~\ref{har:lp} and
Theorem~\ref{T} for the precise statement in the general case of a
discrete group $\Ga$.



In the case of quasi-normed Schatten-von-Neumann classes $\Sch^p$ with
$p<1$, the transfer technique yields a new proof for the following
result of Alexey Alexandrov and Vladimir Peller.

\begin{thm}
  Let~$0<p<1$. The Fourier multiplier $\ph$ is contractive on~$\Ell^p$
  or on~$\Ell^p(\Sch^p)$ if and only if the Schur multiplier~$\Hph$ is
  contractive on~$\Sch^p$ or on~$\Sch^p(\Sch^p)$, if and only if the
  sequence~$\ph$ is the Fourier transform of an atomic measure of the
  form $\sum a_g \delta_{g}$ on~$\T$ with~$\sum\abs{a_g}^p\le 1$.
\end{thm}


The emphasis put on \emph{relative} Schur multipliers motivates the
natural question of how the norm of an elementary Schur multiplier,
that is a rank~1 matrix $(\rh_{r,c})=(x_ry_c)$, gets affected when the
action of~$\rh$ is restricted to matrices with a given support. The
surprising answer is the following.
\begin{thm}
  \label{imps}
  Let $I\subseteq\Row\times\Col$ and consider
  $(x_\row)_{\row\in\Row}$~and~$(y_\col)_{\col\in\Col}$. The relative
  Schur multiplier on~$\Sch^\infty_I$ given by $(x_\row
  y_\col)_{(\row,\col)\in I}$ has norm~$\sup_{(\row,\col)\in
    I}\abs{x_\row y_\col}$.
\end{thm}

Let us finally describe the content of this
article. Section~\ref{sec:tr} is devoted to transfer techniques for
Fourier and Schur multipliers provided by a block matrix Szeg\H{o}
limit theorem. This theorem provides local embeddings of $\Ell^\psi$
into $\Sch^\psi$; Section~\ref{sec:loc} shows how interpolation may be
used to define such embeddings for the scale of $\Ell^p$
spaces. Section~\ref{sec:lac} is devoted to the transfer of lacunary
sets into lacunary matrix patterns; the unconditional constant of a
set $\La$ is related to the size of the sumsets it
contains. Section~\ref{sec:tsp1} deals with Toeplitz Schur multipliers
for $p<1$ and comments on the case $p\ge1$. The Riesz projection and
the Hilbert transform are studied in Section~\ref{sec:rpht}. In
Section~\ref{sec:uap}, the presence of sumsets in a spectrum~$\La$ is
shown to be an obstruction for the existence of completely
unconditional bases for $\Ell^p_\La$. The last section provides a
norm-preserving extension for partially specified rank~1 Schur
multipliers.
 
\begin{NotTer}
  Let~$\T=\{z\in\C:|z|=1\}$. 
  
  Given an index set~$C$ and~$c\in C$, \ $\e_c$~is the sequence
  defined on~$C$ as the indicator function~$\un_{\{c\}}$ of the
  singleton~$\{c\}$, so that $(\e_c)_{c\in C}$~is the canonical
  Schauder basis of the Hilbert space of square summable sequences
  indexed by~$C$, denoted by~$\ell^2_C$. We will use the
  notation~$\ell^2_n=\ell^2_{\{1,2,\dots,n\}}$
  and~$\ell^2=\ell^2_{\N}$.

  Given a product set~$I=\Row\times\Col$ and~$q=(r,c)$, the indicator
  function~$\e_q=\e_{r,c}$ is the \emph{elementary matrix} identified
  with the linear operator from~$\ell^2_\Col$ to~$\ell^2_\Row$ that
  maps~$\e_\col$ on~$\e_\row$ and all other basis vectors on~0. The
  \emph{matrix coefficient} at coordinate~$q$ of a linear operator~$x$
  from~$\ell^2_\Col$ to~$\ell^2_\Row$ is~$x_q=\tr\e_q^*x$ and its
  \emph{matrix representation}
  is~$(x_q)_{q\in\Row\times\Col}=\sum_{q\in\Row\times \Col}x_q\e_q$.
  The \emph{support} or \emph{pattern} of~$x$
  is~$\{q\in\Row\times\Col:x_q\ne0\}$.

  The space of all bounded operators from~$\ell^2_\Col$
  to~$\ell^2_\Row$ is denoted by~$\borne(\ell^2_\Col,\ell^2_\Row)$,
  and its subspace of compact operators is denoted by~$\Sch^\infty$.

  Let $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing function
  vanishing only at~0. The \emph{Schatten-von-Neumann-Orlicz
    class}~$\Sch^\psi$ is the space of those compact operators~$x$
  from~$\ell^2_\Col$ to~$\ell^2_\Row$ such that
  $\tr\psi(|x|/a)<\infty$ for some $a>0$. If $\psi$~is convex, then
  $\Sch^\psi$~is a Banach space for the norm given by
  $\|x\|_{\Sch^\psi}=\inf\{a>0:\tr\psi(|x|/a)\le1\}$. Otherwise,
  $\Sch^\psi$ is a Fr\'echet space for the F-norm given by
  $\|x\|_{\Sch^\psi}=\inf\{a>0:\tr\psi(|x|/a)\le a\}$ (see
  \cite[Chapter~3]{or92}.) This space may also be constructed as the
  noncommutative Lebesgue-Orlicz space~$\Ell^\psi(\tr)$ associated to
  the von~Neumann algebra~$\borne(\ell^2_\Col,\ell^2_\Row)$ endowed
  with the normal faithful semifinite trace~$\tr$.  If $\psi$~is the
  power function~$t\mapsto t^p$, this space is denoted~$\Sch^p$: if
  $p\ge1$, then $\|x\|_{\Sch^p}={(\tr|x|^p)}^{1/p}$; if $p<1$, then
  $\|x\|_{\Sch^p}={(\tr|x|^p)}^{1/(1+p)}$.


  If~$\card\Col=\card\Row=n$, then $\borne(\ell^2_\Col,\ell^2_\Row)$
  identifies with the space of $n\times n$ matrices denoted~$\mat_n$,
  and we write~$\Sch^\psi_n$ for~$\Sch^\psi$. Let
  $(\Row_n\times\Col_n)$~be a sequence of finite sets such that each
  element of~$\Row\times\Col$ eventually is
  in~$\Row_n\times\Col_n$. Then the sequence of operators~$P_n\colon
  x\mapsto\sum_{q\in\Row_n\times \Col_n}x_q\e_q$ tends pointwise to
  the identity on~$\Sch^\psi$.
  
  For~$I\subseteq\Row\times\Col$, we define the space~$\Sch^\psi_I$ as
  the closed subspace of~$\Sch^\psi$ spanned by~$(\e_q)_{q\in I}$: 
  this coincides with the subspace of those~$x\in \Sch^\psi$ whose
  support is a subset of~$I$.
  
  A \emph{relative Schur multiplier} on~$\Sch^\psi_I$ is a
  sequence~$\rh=(\rh_q)_{q\in I}\in\C^I$ such that the associated
  \emph{Schur multiplication operator}~$\Schur_\rh$ defined
  by~$\e_q\mapsto\rh_q\e_q$ for~$q\in I$ is bounded on~$\Sch^\psi_I$.
  The norm~$\|\rh\|_{\Schur(\Sch^\psi_I)}$ of~$\rh$ is defined as the
  norm of~$\Schur_\rh$.  This norm is the supremum of the norm of its
  restrictions to finite rectangle sets~$\Row'\times\Col'$.  We used
  \cite{pi98,pi01} as a reference.
  
  Let $\Ga$~be a discrete group with identity~$\id$. The \emph{reduced
    $\Cont^*$-algebra} of~$\Ga$ is the closed subspace spanned by the
  left translations~$\la_\ga$ (the linear operators defined
  on~$\ell^2_\Ga$ by~$\la_\ga\e_\beta=\e_{\ga\beta}$)
  in~$\borne(\ell^2_\Ga)$; we denote it by~$\Cont$, set in roman
  type. The \emph{von~Neumann algebra} of~$\Ga$ is its
  $\textrm{weak}^*$ closure, endowed with the normal faithful
  normalised finite trace~$\tau$ defined by~$\tau(x)=x_{\id,\id}$; we
  denote it by~$\vn$.  Let $\psi\colon\R^+\to\R^+$ be a continuous
  nondecreasing function vanishing only at~0: we define the
  noncommutative Lebesgue-Orlicz space~$\Ell^\psi$ of $\Ga$ as the
  completion of $\vn$ with respect to the norm given by
  $\|x\|_{\Ell^\psi}=\inf\{a>0:\tau(\psi(|x|/a))\le1\}$ if $\psi$~is
  convex and with respect to the $F$-norm given
  by$\|x\|_{\Ell^\psi}=\inf\{a>0:\tau(\psi(|x|/a))\le a\}$ otherwise.
  If $\psi$~is the power function~$t\mapsto t^p$, this space is
  denoted~$\Ell^p$: if $p\ge1$, then
  $\|x\|_{\Ell^p}={\tau(|x|^p)}^{1/p}$; if $p<1$, then
  $\|x\|_{\Ell^p}={\tau(|x|^p)}^{1/(1+p)}$.  The \emph{Fourier
    coefficient} of~$x$ at~$\ga$
  is~$x_\ga=\tau(\la_\ga^*x)=x_{\ga,\id}$ and its \emph{Fourier
    series} is~$\sum_{\ga\in\Ga}x_\ga\la_\ga$. The \emph{spectrum} of
  an element~$x$ is~$\{\ga\in\Ga:x_\ga\ne0\}$. Let $X$~be the
  $\Cont^*$-algebra~$\Cont$ or the space~$\Ell^\psi$ and
  let~$\La\subseteq\Ga$: then we define $X_\La$~as the
  closed subspace of~$X$ spanned by the~$\la_\ga$ with~$\ga\in\La$. We
  skip the general question for which spaces~$X$ this coincides with
  the subspace of those~$x\in X$ whose spectrum is a subset of~$\La$,
  but note that this is the case if $\Ga$ is an amenable group (or if
  $\vn$~has the QWEP by \cite[Theorem~4.4]{jr03}) and $\psi$~is the
  power function~$t\mapsto t^p$; note also that our definition of
  $X_\La$ makes it a subspace of the \emph{heart} of $X$:  if $x\in X_\La$, then $\tau(\psi(|x|/a))$ is finite for all $a>0$.

  A \emph{relative Fourier multiplier} on~$X_\La$ is a
  sequence~$\ph=(\ph_\ga)_{\ga\in\La}\in\C^\La$ such that the
  associated Fourier multiplication operator~$\Fourier_\ph$ defined
  by~$\la_\ga\mapsto\ph_\ga\la_\ga$ for~$\ga\in\La$ is bounded
  on~$X_\La$. The norm~$\|\ph\|_{\Fourier(X_\La)}$ of~$\ph$ is defined
  as the norm of~$\Fourier_\ph$. 
  Fourier multipliers on the whole of the $\Cont^*$-algebra $\Cont$
  are also called \emph{multipliers of the Fourier algebra ${\mathrm
      A}(\Ga)$} (which may be identified with $\Ell^1$); they form the
  set $\Fourier(\mathrm{A}(\Ga))$.

  The space~$\Sch^\psi(\Sch^\psi)$ is the space of those compact
  operators~$x$ from~$\ell^2\otimes\ell^2_\Col$
  to~$\ell^2\otimes\ell^2_\Row$ such that
  $\|x\|_{\Sch^\psi(\Sch^\psi)}=\inf\{a:\tr\otimes\tr\psi(|x|/a)\le1\}$:
  it is the noncommutative Lebesgue-Orlicz
  space~$\Ell^\psi(\tr\otimes\tr)$ associated to the von~Neumann
  algebra~$\borne(\ell^2)\otimes\borne(\ell^2_C,\ell^2_R)$. One may
  think of~$\Sch^\psi(\Sch^\psi)$ as the $\Sch^\psi$-valued
  Schatten-von-Neumann class: we define the matrix coefficient of~$x$
  at~$q$ by~$x_q = (\Id_{\Sch^\psi}\otimes\tr)\allowbreak
  \bigl((\Id_{\ell^2}\otimes\e_q^*)x\bigr)\in\Sch^\psi$ and its matrix
  representation by $\sum_{q\in\Row\times\Col}x_q\tens\e_q$. The
  support of~$x$ and the subspace~$\Sch^\psi_I(\Sch^\psi)$ are defined
  in the same way as~$\Sch^\psi_I$.
  
  Similarly, the space~$\Ell^\psi(\tr\otimes\tau)$ is the noncommutative
  Lebesgue-Orlicz space associated to the von~Neumann algebra~$\borne(\ell^2)\otimes\vn=\Ell^\infty(\tr\otimes\tau)$. One may think of~$\Ell^\psi(\tr\otimes\tau)$ as the $\Sch^\psi$-valued noncommutative
  Lebesgue space: we define the Fourier coefficient of~$x$ at~$\ga$
  by~$x_\ga=(\Id_{\Sch^\psi}\otimes\tau)\bigl((\Id_{\ell^2}\otimes\la_\ga^*)x\bigr)\in\Sch^\psi$
  and its Fourier series
  by~$\sum_{\ga\in\Ga}x_\ga\tens\la_\ga$; the spectrum
  of~$x$ is defined accordingly. The
  subspace~$\Ell^\psi_\La(\tr\otimes\tau)$ is the closed subspace
  of~$\Ell^\psi(\tr\otimes\tau)$ spanned by
  the~$x\otimes\la_\ga$ with~$x\in\Sch^\psi$
  and~$\ga\in\La$. 
  
  An operator~$T$ on~$\Sch^\psi_I$ is \emph{bounded on
    $\Sch^\psi_I(\Sch^\psi)$} if the linear operator $\Id_{\Sch^\psi}\otimes T$
  defined by~$x\tens y\mapsto x\tens T(y)$ for~$x\in \Sch^\psi$ and $y$
  in~$\Sch^\psi_I$ on finite tensors extends to a bounded
  operator~$\Id_{\Sch^\psi}\otimes T$ on~$\Sch^\psi_I(\Sch^\psi)$.  The norm of
  a Schur multiplier~$\rh$ on $\Sch^\psi_I(\Sch^\psi)$ is defined as the
  norm of~$\Id_{\Sch^\psi}\otimes\Schur_\rh$.
  Similar definitions hold for an operator~$T$ on~$\Ell^\psi_\La$; the
  norm of a Fourier multiplier~$\ph$ on~$\Ell^\psi_\La(\tr\otimes\tau)$
  is the norm of $\Id_\Sch^\psi\otimes\Fourier_\ph$
  on~$\Ell^\psi_\La(\tr\otimes\tau)$.

  Let $\psi$~be the power function~$t\mapsto t^p$ with $p\ge1$: the
  norms on $\Sch^p(\Sch^p)$ and $\Ell^p(\tr\otimes\tau)$ describe the
  canonical operator space structure on $\Sch^p$~and~$\Ell^p$,
  respectively: see \cite[Corollary~1.4]{pi98}; we should rather use
  the notation $\Sch^p[\Sch^p]$ and $\Sch^p[\Ell^p]$. This explains
  the following terminology.  An operator~$T$ on~$\Sch^p_I$ is
  \emph{completely bounded} (c.b.~for short) if $\Id_{\Sch^p}\otimes
  T$~is bounded on~$\Sch^p_I(\Sch^p)$; the norm
  of~$\Id_{\Sch^p}\otimes T$ is the \emph{complete norm} of~$T$
  (compare \cite[Lemma~1.7]{pi98}.)  The complete
  norm~$\|\rh\|_{\Schur_\cb(\Sch^p_I)}$ of a Schur multiplier~$\rh$ is
  defined as the complete norm of~$\Schur_\rh$. Note that the complete
  norm of a Schur multiplier~$\rh$ on~$\Sch^\infty_I$ is equal to its
  norm \cite[Theorem~3.2]{pps89}: \
  $\|\rh\|_{\Schur_\cb(\Sch^\infty_I)}=\|\rh\|_{\Schur(\Sch^\infty_I)}$.
  The complete norm~$\|\ph\|_{\Fourier_\cb(\Ell^p_\La)}$ of a Fourier
  multiplier~$\ph$ is defined as the complete norm of~$\Fourier_\ph$.
  The complete norm of an operator~$T$ on~$\Cont_\La$ is the norm
  of~$\Id_{\Sch^\infty}\otimes T$ on the subspace
  of~$\Sch^\infty\otimes\Cont$ spanned by the~$x\otimes\la_\ga$
  with~$x\in\Sch^\infty$ and~$\ga\in\La$; in the case $\La=\Ga$, $\ph$
  is also called a \emph{c.b.\ multiplier of the Fourier algebra
    ${\mathrm A}(\Ga)$} and one writes $\ph\in\Fourier_{\cb}(\mathrm
  A(\Ga))$; if $\Ga$~is amenable, the complete norm of a Fourier
  multiplier~$\ph$ on~$\Cont_\La$ is equal to its norm
  \cite[Corollary~1.8]{dh85}: \
  $\|\ph\|_{\Fourier_\cb(\Cont_\La)}=\|\ph\|_{\Fourier(\Cont_\La)}$.

  An element whose norm is at most~1 is
  \emph{contractive}, and if its complete norm is at most~1, it is
  \emph{completely} contractive.


  If $\Ga$~is abelian, let $G$~be its dual group and endow it with its
  unique normalised Haar measure~$m$: then the Fourier transform
  identifies the $\Cont^*$-algebra~$\Cont$ as the space of continuous functions on~$G$, \
  $\Ell^\infty$ as the space of classes of bounded measurable
  functions on~$(G,m)$, \ $\Ell^\psi$ as the Lebesgue-Orlicz space of classes
  of~$\psi$-integrable functions on~$(G,m)$, \ $\tau(x)$
  as~$\int_Gx(g)\,\mathrm dm(g)$, \ $\Ell^\psi(\tr\otimes\tau)$ as the $\Sch^\psi$-valued Lebesgue-Orlicz space $\Ell^\psi(\Sch^\psi)$  and $x_\ga$ as~$\hat x(\ga)$.

\end{NotTer}

\section{Transfer between Fourier and Schur multipliers}\label{sec:tr}


Let $\La$~be a subset of a discrete group~$\Ga$ and let $\ph$~be a
relative Fourier multiplier on~$\Cont_\La$, the closed subspace spanned by $(\la_\ga)_{\ga\in\La}$ in the reduced $\Cont^*$-algebra of $\Ga$. Let $x\in\Cont_\La$: the
matrix of~$x$ is \emph{constant down the diagonals} in the sense that
for every~$(r,c)\in\Ga\times\Ga$, \
$x_{r,c}=x_{rc^{-1},\id}=x_{rc^{-1}}$; we say that $x$~is a
\emph{Toeplitz} operator on~$\ell^2_\Ga$. Furthermore, the matrix of
the Fourier product~$\Fourier_\ph x$ of~$\ph$ with~$x$ is given
by~$(\Fourier_\ph x)_{r,c}=\ph_{rc^{-1}}x_{r,c}$. This shows that if we
set~$\HLa=\{(r,c)\in\Ga\times\Ga:\row\col^{-1}\in
\La\}$~and~$\Hph_{r,c}=\ph_{\row\col^{-1}}$, then $\Fourier_\ph x$~is the Schur
product $\Schur_\Hph x$ of~$\Hph$ with~$x$.  We have transferred the
Fourier multiplier~$\ph$ into the Schur multiplier~$\Hph$: this shows
at once that the norm of the Fourier multiplier~$\ph$ on~$\Cont_\La$
is the norm of the Schur multiplier~$\Hph$ on the subspace of Toeplitz
elements of~$\borne(\ell^2_\Ga)$ with support in~$\HLa$, and that the
same holds for complete norms.

We shall now give us the means to generalise this identification to
the setting of Lebesgue-Orlicz spaces $\Ell^\psi$: we shall bypass the main obstacle, that $\Ell^\psi$~may not
be considered as a subspace of~$\Sch^\psi$, by the Szeg\H{o} limit
theorem as stated by Erik B\'edos \cite[Theorem~10]{be97}.

As we want to compute complete norms of multipliers, we shall
generalise the Szeg\H{o} limit theorem to the block matrix case, not
considered in \cite{be97}.  Let us first recall the scalar
case. Consider a discrete amenable group~$\Ga$: it admits a
\emph{F{\o}lner averaging net of sets} $(\Ga_\iota)$, that is,
\begin{itemize}
\item each~$\Ga_\iota$ is a finite subset of~$\Ga$;
\item
  $\card(\ga\Ga_\iota\Delta\Ga_\iota)=o(\card\Ga_\iota)$
  for each~$\ga\in\Ga$.
\end{itemize}
Each set~$\Ga_\iota$ corresponds to the orthogonal
projection~$p_\iota$ of~$\ell^2_\Ga$ onto its
$(\card\Ga_\iota)$-dimensional subspace of sequences supported
by~$\Ga_\iota$. The \emph{truncate} of a selfadjoint
operator~$y\in\borne(\ell^2_\Ga)$ with respect to~$\Ga_\iota$ is
$y_\iota=p_\iota yp_\iota$: it has $\card\Ga_\iota$
eigenvalues~$\alpha_j$, counted with multiplicities, and its
\emph{normalised counting measure of eigenvalues} is
\begin{equation*}
  \mu_\iota=\frac1{\card\Ga_\iota}\sum_{j=1}^{\card\Ga_\iota}\delta_{\alpha_j}.
\end{equation*}
If $y$~is a Toeplitz operator, that is, if $y\in\vn$, Erik B\'edos
\cite[Theorem~10]{be97} proves that $(\mu_\iota)$ converges
$\textrm{weak}^*$ to the \emph{spectral measure} of~$y$ with respect
to~$\tau$, which is the unique Borel probability measure~$\mu$ on~$\R$
such that
\begin{equation*}
  \tau(\psi(y))=\int_\R\psi(\alpha)\mathrm d\mu(\alpha)
\end{equation*}
for every continuous function~$\psi$ on~$\R$ that tends to zero at
infinity. If $\Ga$~is abelian, then $y$~may be identified as the
class of a real-valued bounded measurable function on the group~$G$
dual to~$\Ga$ and $\mu$~is the distribution of~$y$.

The matrix Szeg\H{o} limit theorem is the analogue of this result for
selfadjoint elements $y\in\mat_n\otimes\vn$, whose \emph{$\mat_n$-valued spectral measure}~$\mu$ is defined by 
\begin{equation*}
  \int_\R\psi(\alpha)\mathrm d\mu(\alpha)=\Id_{\ell^2_n}\otimes\tau(\psi(y)).
\end{equation*}
The orthogonal projection~$\tilde
p_\iota=\Id_{\ell^2_n}\otimes p_\iota$ defines the truncate~$y_\iota=\tilde p_\iota y\tilde
p_\iota\in\mat_n\otimes\borne(\ell^2_{\Ga_\iota})$, and the \emph{$\mat_n$-valued normalised counting
measure of eigenvalues}~$\mu_\iota$ by
\begin{equation*}
  \int_\R\psi(\alpha)\mathrm d\mu_\iota(\alpha)=\Id_{\ell^2_n}\otimes\frac{\tr}{\card\Ga_\iota}(\psi(y_\iota)).
\end{equation*}

\begin{thm}[Matrix Szeg\H{o} limit theorem]
  \label{sz}
  Let $\Ga$~be a discrete amenable group and let $(\Ga_\iota)$~be a F{\o}lner averaging net for~$\Ga$. Let $y$~be a selfadjoint
  element of~$\mat_n\otimes\vn$. The net~$(\mu_\iota)$ of~$\mat_n$-valued normalised
  counting measures of eigenvalues of the truncates of~$y$ with respect
  to~$\Ga_\iota$ converges in the $\textrm{weak}^*$ topology to
  the spectral measure of~$y$:
  \begin{equation*}
    \int_\R\psi(\alpha)\mathrm d\mu_\iota(\alpha)\to\Id_{\ell^2_n}\otimes\tau(\psi(y))
  \end{equation*}
  for every continuous function~$\psi$ on~$\R$ that tends to zero at
  infinity.
\end{thm}

\begin{proof}[Sketch of proof]
  Let us first suppose that~$y\in\mat_n\otimes\Cont$: we may suppose that
  $y=\sum_{\ga\in\Ga}y_\ga\tens\la_\ga$ with only a
  finite number of the~$y_\ga\in\mat_n$ nonzero: the~$\mat_n$-valued matrix of~$y_\iota$ for the canonical basis of~$\ell^2_{\Ga_\iota}$ is~$(y_{rc^{-1}})_{(r,c)\in\Ga_\iota\times\Ga_\iota}$. It
  suffices to prove that
  \begin{equation}
    \label{szego}
    \Id\otimes\frac\tr{\#\Ga_\iota}(y_\iota^k)\to\Id\otimes\tau(y^k)
  \end{equation}
  for every~$k$. This is trivial if~$k=0$. If~$k=1$, then
  \begin{equation*}
    \Id\otimes\frac\tr{\#\Ga_\iota}(y_\iota)
    =\frac1{\#\Ga_\iota}\sum_{\col\in\Ga_\iota}y_{\col,\col}
    =\Id\otimes\tau(y)
  \end{equation*}
  as~$y_{\col,\col}=y_{\col\col^{-1}}=y_\id$. If~$k\ge2$, the same
  formula holds with~$y^k$ instead of~$y$:
  \begin{equation*}
    \Id\otimes\tau(y^k)=\Id\otimes\frac\tr{\#\Ga_\iota}
    (\tilde p_\iota y^k\tilde p_\iota),
  \end{equation*}
  so that we wish to prove
  \begin{equation*}
    \Id\otimes\tr(\tilde p_\iota y^k\tilde p_\iota
    -{(\tilde p_\iota y\tilde p_\iota)}^k)
    =o(\card\Ga_\iota).
  \end{equation*}
  Note that 
  \begin{equation*}
    \bignorm{\Id\otimes\tr\bigl(\tilde p_\iota y^k\tilde p_\iota-{(\tilde p_\iota y\tilde p_\iota)}^k\bigr)}_{\Sch^1_n}
      \le\|\tilde p_\iota y^k\tilde p_\iota-{(\tilde p_\iota y\tilde p_\iota)}^k\|_{\Sch^1(\Sch^1_n)}.
    \end{equation*}
Lemma~5 in \cite{be97} provides the following estimate: as
  \begin{equation*}
    \tilde p_\iota y^k\tilde p_\iota-{(\tilde p_\iota y\tilde p_\iota)}^k=\tilde p_\iota y^{k-1}(y\tilde p_\iota-\tilde p_\iota y\tilde p_\iota)+(\tilde p_\iota y^{k-1}\tilde p_\iota-{(\tilde p_\iota y\tilde p_\iota)}^{k-1})y\tilde p_\iota,
  \end{equation*}
  an induction yields 
  \begin{equation*}
    \|\tilde p_\iota y^k\tilde p_\iota-{(\tilde p_\iota y\tilde p_\iota)}^k\|_{\Sch^1(\Sch^1_n)}
      \le(k-1)\|y\|_{\mat_n\otimes\vn}^{k-1}\|y\tilde p_\iota-\tilde p_\iota y\tilde p_\iota\|_{\Sch^1(\Sch^1_n)}.
  \end{equation*}
  It suffices to consider the very last norm for each term~$y_\ga\tens\la_\ga$ of~$y$: let~$h\in\ell^2_n$ and~$\beta\in\Ga$; as
  \begin{equation*}
    \bigl((y_\ga\tens\la_\ga)\tilde p_\iota-\tilde p_\iota(y_\ga\tens\la_\ga)\tilde p_\iota\bigr)(h\otimes\e_\beta)
    =
    \begin{cases}
      y_\ga(h)\e_{\ga\beta}&\text{if $\beta\in\Ga_\iota$ and $\ga\beta\notin\Ga_\iota$}\\
      0&\text{otherwise,}
    \end{cases}
  \end{equation*}
  the definition of a F{\o}lner averaging net yields
  \begin{equation*}
    \|(y_\ga\tens\la_\ga)\tilde p_\iota
    -\tilde p_\iota (y_\ga\tens\la_\ga)\tilde p_\iota\|
    _{\Sch^1(\Sch^1_n)}
    \le\#(\Ga_\iota\setminus\ga^{-1}\Ga_\iota)\|y_\ga\|_{\Sch^1_n}
    =o(\card\Ga_\iota).
  \end{equation*}
  An approximation argument as in \cite[proof of Proposition~4]{be97}
  permits to conclude for~$y\in\vn$.
\end{proof}

Let us now describe and prove the $\Ell^\psi$ version of the transfer
described at the beginning of this section. 
\begin{lem}
  \label{trprp}
  Let $\Ga$~be a discrete amenable group and~$p>0$. Let
  $\La\subseteq\Ga$ and~$\ph\in\C^\La$. Consider the associated Toeplitz
  set~$\HLa=\{(r,c)\in\Ga\times\Ga:\row\col^{-1}\in\La\}$ and the
  Toeplitz matrix defined by~$\Hph_{r,c}=\ph_{\row\col^{-1}}$. Let
  $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing function vanishing only at~0.
  \begin{enumerate}
  \item The norm of the relative Fourier multiplier~$\ph$
    on~$\Ell^\psi_\La$ is bounded by the norm of the relative
    Schur multiplier~$\Hph$ on~$\Sch^\psi_{\mkern-3mu\HLa}$.
  \item The norm of the relative Fourier multiplier~$\ph$ on
    $\Ell^\psi_\La(\tr\otimes\tau)$ is bounded by the norm of the
    relative Schur multiplier~$\Hph$
    on~$\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)$.
  \end{enumerate}
\end{lem}
Theorem~\ref{trans2}$\,(a)$ below will provide the
full picture of case~$(b)$: both norms are in fact equal. This is not so in case~$(a)$: see Remark~\ref{b-cb}. 
\begin{proof}
  A Toeplitz matrix has the form~$(x_{rc^{-1}})_{(r,c)\in \HLa}$. Our
  definition of the space~$\Ell^\psi_\La$ (cf.~section on Notation and terminology) ensures that
  we may suppose that only a finite number of the~$x_\ga$ are nonzero
  for the computation of the norm of~$\ph$. Then
  $(x_{rc^{-1}})_{(r,c)\in \HLa}$~is the matrix of the
  operator~$x=\sum_{\ga\in \La}x_\ga\la_\ga$ for the canonical basis
  of~$\ell^2_\Ga$.

  Let~$y=x^*x$ and let us use the notation of Theorem~\ref{sz}. Let
  $\tilde\psi$~be a continuous function with compact support such that
  $\tilde\psi(t)=\psi(t)$ on~$[0,\|x\|^2]$.
  
  $(a)$. By Szeg\H{o}'s limit theorem (as given in
  \cite[Theorem~1]{be97}),
  \begin{equation*}
\frac1{\#\Ga_\iota}\tr\psi(p_\iota yp_\iota)=
\frac1{\#\Ga_\iota}\tr\tilde\psi(p_\iota yp_\iota)
    \to\tau(\tilde\psi(y))=\tau(\psi(y))
  \end{equation*}
  Let us describe how $\Hph$ acts on~$xp_\iota$. Schur multiplication
  with~$\Hph$ transforms the matrix of~$xp_\iota$, that is the
  truncated Toeplitz matrix~$(x_{rc^{-1}})_{(r,c)\in
    \HLa\cap\Ga\times\Ga_\iota}$, into the matrix~$(\Hph_{rc^{-1}}x_{rc^{-1}})_{(r,c)\in
    \HLa\cap\Ga\times\Ga_\iota}$, so that it transforms~$xp_\iota$
  into~$(\Fourier_\ph x)p_\iota$.

  $(b)$. Combine the argument in~$(a)$ with the matrix Szeg\H{o}
  limit theorem.
\end{proof}
In the case of a finite abelian group, no limit theorem is needed:
this case has been considered in~\cite[Proposition~2.5$\,(b)$]{ne06}.

\begin{rem}
  Our technique proves in fact that the norm of a Fourier multiplier
  is the upper limit of the norm of the corresponding relative Schur multipliers on
  subspaces of truncated Toeplitz matrices. We ignore whether it is
  actually their supremum.
\end{rem}

The following well-known argument has been used (first in~\cite{bf84},
see~\cite[Proposition~D.6]{bo08}) to show that the complete norm of the
Fourier multiplier~$\ph$ on~$\vn_\La$ bounds the complete norm of
the Schur multiplier~$\Hph$ on $\Sch^\infty_{\mkern-3mu\HLa}$, so that we have in full
generality
$\|\ph\|_{\Fourier_\cb(\Cont_\La)}=\|\Hph\|_{\Schur_\cb(\Sch^\infty_{\mkern-3mu\HLa})}$.
This argument permits to strengthen Lemma~\ref{trprp}$\,(b)$.

\begin{lem}
  \label{trans1}
  Let $\Ga$~be a discrete group and let $\Row$~and~$\Col$ be subsets
  of~$\Ga$. To~$\La\subseteq\Ga$
  associate~$\HLa=\{(r,c)\in\Row\times\Col:\row\col^{-1}\in \La\}$;
  given~$\ph\in\C^\La$ define~$\Hph\in\C^\HLa$
  by~$\Hph_{r,c}=\ph_{\row\col^{-1}}$.  Let $\psi\colon\R^+\to\R^+$ be
  a continuous nondecreasing function vanishing only at~0. The norm of
  the relative Schur multiplier~${\Hph}$
  on~$\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)$ is bounded by the norm of
  the relative Fourier multiplier~$\ph$
  on~$\Ell^\psi_\La(\tr\otimes\tau)$.
\end{lem}
\begin{proof}
  We adapt the argument in~\cite[Lemma~8.1.4]{pi98}. Let~$x_q\in
  \Sch^\psi$, of which only a finite number are nonzero. The space~$\Ell^\psi(\tr\otimes\tr\otimes\tau)$ is a left and right
  $\Ell^\infty(\tr\otimes\tr\otimes\tau)$-module
  and~$\sum_{\ga\in\Ga}\e_{\ga\ga}\tens\la_\ga$
  is a unitary in~$\Ell^\infty(\tr\otimes\tau)$, so that
  \begin{align*}
    \Bigl\|\sum_{q\in \HLa}x_q\tens\e_q\Bigr\|_{\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)}
      &=
      \Bigl\|\Bigl(\Id\otimes\sum_{r\in\Row}\e_{r,r}\tens\la_\row\Bigr)
      \Bigl(\sum_{q\in \HLa}x_q\tens\e_q\tens\la_\id\Bigr)
      \Bigl(\Id\otimes\sum_{c\in\Col}\e_{c,c}\tens\la_\col^*\Bigr)\Bigr\|_{\Ell^\psi(\tr\otimes\tr\otimes\tau)}\\
      &= \biggl\| \sum_{(r,c)\in \HLa}
      x_{r,c}\tens\e_{r,c}\tens\la_{\row\col^{-1}}\biggr\|_{\Ell^\psi(\tr\otimes\tr\otimes\tau)}\\
      &= \biggl\| \sum_{\ga\in \La}
      \biggl(\sum_{\row\col^{-1}=\ga}
      x_{r,c}\tens\e_{r,c}\biggr)\tens\la_\ga\biggr\|_{\Ell^\psi_\La(\tr\otimes\tr\otimes\tau)}.
  \end{align*}
  This yields an isometric embedding of~$\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)$ in~$\Ell^\psi_\La(\tr\otimes\tr\otimes\tau)$. As $\Sch^\psi(\Sch^\psi)$~is the
  Schatten-von-Neumann-Orlicz class for the Hilbert space~$\ell^2\otimes\ell^2_\Ga$ which may be identified with~$\ell^2$,
  \begin{multline}
    \Bigl\| \sum_{q\in \HLa} x_q\tens\Hph_q\e_q \Bigr\|_{\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)}
    = \biggl\| \sum_{\ga\in \La} \biggl(
    \sum_{\row\col^{-1}=\ga} x_{r,c}\tens\e_{r,c}
    \biggr)\tens\ph_\ga     \la_\ga \biggr\|_{\Ell^\psi_\La(\tr\otimes\tr\otimes\tau)}\\
    \le \|\Id_{\Sch^\psi}\otimes\Fourier_\ph\| \Bigl\| \sum_{q\in \HLa} x_q\tens\e_q
    \Bigr\|_{\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)}.\tag*{\qedhere}
  \end{multline}
\end{proof}

\begin{rem}
  \label{transg}
  The proof of Lemma~\ref{trans1} shows also the following
  transfer.  Let $(r_i)$ and $(c_j)$ be sequences in~$\Ga$,
  consider~$\bLa=\{(i,j)\in\N\times\N:r_ic_j\in\La\}$ and
  define~$\bph\in\C^\bLa$ by~$\bph(i,j)=\ph(r_ic_j)$. Then the norm of
  the relative Schur multiplier~${\bph}$
  on~$\Sch^\psi_{\bLa}(\Sch^\psi)$ is bounded by the norm of the
  relative Fourier multiplier~$\Id_{\Sch^\psi}\otimes\Fourier_\ph$
  on~$\Ell^\psi_\La(\tr\otimes\tau)$ (compare with
  \cite[Theorem~6.4]{pi01}.) In particular, if the $r_ic_j$ are
  pairwise distinct, this permits to transfer every Schur multiplier,
  not just the Toeplitz ones. See \cite[Section~11]{ne06} for
  applications of this transfer.
\end{rem}

Here is the announced strengthening of Lemma~\ref{trprp}.

\begin{thm}\label{trans2}
  Let $\Ga$~be a discrete amenable group. Let
  $\La\subseteq\Ga$ and $\ph\in\C^\La$. Consider the
  associated Toeplitz set~$\HLa=\{(r,c)\in\Ga\times\Ga:\row\col^{-1}\in
  \La\}$ and the Toeplitz matrix defined by~$\Hph_{r,c}=\ph_{\row\col^{-1}}$.
  \begin{enumerate}
  \item Let $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing
    function vanishing only at~0. The norm of the relative Fourier
    multiplier~$\ph$
    on~$\Ell^\psi_\La(\tr\otimes\tau)$ and the norm of the
    relative Schur multiplier~${\Hph}$
    on~$\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)$ are equal.
  \item Let $p\ge1$. The complete norm of the relative Fourier
    multiplier~$\ph$ on~$\Ell^p_\La$ and the complete norm of the
    relative Schur multiplier~$\Hph$ on~$\Sch^p_{\mkern-3mu\HLa}$ are
    equal:
    \begin{equation*}
      \|\ph\|_{\Fourier_\cb(\Ell^p_\La)}=\|\Hph\|_{\Schur_\cb(\Sch^p_{\mkern-3mu\HLa})}.
    \end{equation*}
  \item The norm of the relative Fourier multiplier~$\ph$ on~$\Cont_\La$, its complete norm, the norm of the relative Schur
    multiplier~$\Hph$ on~$\Sch^\infty_{\mkern-3mu\HLa}$ and its complete norm are equal:
    \begin{equation*}
      \|\ph\|_{\Fourier(\Cont_\La)}
      =\|\ph\|_{\Fourier_\cb(\Cont_\La)}
      =\|\Hph\|_{\Schur_\cb(\Sch^\infty_{\mkern-3mu\HLa})}
      =\|\Hph\|_{\Schur(\Sch^\infty_{\mkern-3mu\HLa})}.
    \end{equation*}
\item Suppose that $\La=\Ga$. The norm of the Fourier algebra
  multiplier~$\ph$, its complete norm, the norm of the Schur
  multiplier~$\Hph$ on~$\Sch^\infty$ and its complete norm are
  equal:
    \begin{equation*}
      \|\ph\|_{\Fourier(\mathrm{A}(\Ga))}
      =\|\ph\|_{\Fourier_\cb(\mathrm A(\Ga))}
      =\|\Hph\|_{\Schur_\cb(\Sch^\infty)}
      =\|\Hph\|_{\Schur(\Sch^\infty)}.
    \end{equation*}
  \end{enumerate}
\end{thm}
\begin{proof}
  Combine Proposition~\ref{trprp}$\,(b)$ with
  Lemma~\ref{trans1}. Recall that if $\Ga$~is amenable, the
  norm of a Fourier multiplier~$\ph$ on~$\Cont_\La$ is
  equal to its complete norm \cite[Corollary~1.8]{dh85} and that the complete norm of a Schur
  multiplier~$\Hph$ on~$\Sch^\infty_{\mkern-3mu\HLa}$ is equal to its norm
  \cite[Theorem~3.2]{pps89}.
\end{proof}



\section{\texorpdfstring{Local embeddings of~$\Ell^p$ into~$\Sch^p$}{Local embeddings of Lp into Sp}}\label{sec:loc}




\noindent The proof of Lemma~\ref{trprp} can be interpreted as an
embedding of $\Ell^\psi$ into an ultraproduct of finite-dimensional
spaces $\Sch^\psi_n$ that intertwines Fourier and Toeplitz Schur
multipliers. If we restrict ourselves to power functions $\psi\colon
t\mapsto t^p$ with $p\ge 1$, such embeddings are well known and the
proof of Lemma~\ref{trprp} does not need the full strength of the
Matrix Szeg\H{o} limit theorem but only the existence of such
embeddings. In this section, we explain two ways to obtain them by
interpolation.


The first way is to extend the classical result that the reduced
$\Cont^*$-algebra~$\Cont$ of a discrete group $\Ga$ has the completely contractive
approximation property if $\Gamma$ is amenable. We follow the approach
of \cite[Theorem~2.6.8]{bo08}.  Let $\Gamma$ be a discrete amenable
group and $\Gamma_\iota$ be a F{\o}lner averaging net of sets. As
above, we denote by $p_\iota$ the orthogonal projection from
$\ell^2_\Gamma$ to $\ell^2_{\Gamma_\iota}$. Define the
compression~$\ph_\iota$ and the embedding $\psi_\iota$ by
\begin{equation}
  \label{phipsi}
  \begin{aligned}[t]
    \ph_\iota\colon\Cont&\to\borne(\ell^2_{\Gamma_\iota})\\
    x&\mapsto p_\iota x p_\iota
  \end{aligned}
  \quad\text{and}\quad
  \begin{aligned}[t]
    \psi_\iota\colon\borne(\ell^2_{\Gamma_\iota})&\to\Cont\\
    \e_{r,c}&\mapsto(1/\card \Gamma_\iota)\la_r\la_{c^{-1}}.
  \end{aligned}
\end{equation}
If we endow $\borne(\ell^2_{\Gamma_\iota})$ with the normalised trace,
these maps are unital completely positive, trace preserving (and
normal) and the net $(\psi_\iota\ph_\iota)$ converges pointwise to the
identity of $\Cont$. One can therefore extend them by interpolation to
completely positive contractions on the respective noncommutative
Lebesgue spaces: recall that $\Ell^p(\borne(\ell^2_{\Gamma_\iota}),
(1/\card\Gamma_\iota)\tr)$ is
${(\card\Gamma_\iota)}^{-1/p}\Sch^p_{\card \Gamma_\iota}$. We get a
net of complete contractions
\begin{equation*}
  \tilde \ph_\iota \colon  \Ell^p\to {(\card\Gamma_\iota)}^{-1/p}\Sch^p_{\card \Gamma_\iota}
  \quad\text{and}\quad
  \tilde \psi_\iota\colon {(\card\Gamma_\iota)}^{-1/p}\Sch^p_{\card \Gamma_\iota} \to \Ell^p,
\end{equation*}
such that $(\tilde\psi_\iota\tilde\ph_\iota)$ converges pointwise
to the identity of $\Ell^p$. Moreover, the definitions~\eqref{phipsi}
show that these maps also intertwine Fourier and Toeplitz Schur
multipliers.

\begin{rem}
  This approach is more canonical as it allows to extend the transfer
  to the vector-valued spaces in the sense of \cite[Chapter
  3]{pi98}. Recall that for any hyperfinite semifinite von Neumann
  algebra $M$ and any operator space $E$, one can define $\Ell^p(M,E)$:
  for $p=\infty$, this space is defined as $M\tens_{\min}E$; for
  $p=1$, this space is defined as $M_*^{\mathrm{op}}\hat\tens E$; these spaces form an
  interpolation scale for the complex method when $1\le p\le
  \infty$. For us, $M$ will be $\borne(\ell^2)$ or the group von
  Neumann algebra $\vn$. As the maps $\psi_\iota$ and $\ph_\iota$ are unital
  completely positive and trace preserving and normal, they define
  simultaneously complete contractions on $M$ and $M_*$. By
  interpolation, the maps $\psi_\iota\tens \Id_E$ and $\ph_\iota\tens
  \Id_E$ are still complete contractions on the spaces $\Ell_p(E)$ and
  $\Sch^p[E]$. Let $\ph\colon \Ga\to \C$: the transfer shows that the
  norm of $\Id_E\otimes\Fourier_\ph$ on~$\Ell^p(E)$ is bounded by the
  norm of $\Id_E\otimes\Schur_\Hph$ on~$\Sch^p[E]$, and that their
  complete norms coincide. In formulas,
  \begin{gather*}
    \|\Id_E\tens\Fourier_\ph\|_{\borne(\Ell^p(E))}
    \le\|\Id_E\tens\Schur_\Hph\|_{\borne(\Sch^p[E])},\\
    \|\Id_E\tens\Fourier_\ph\|_{\cb(\Ell^p(E))}
    =\|\Id_E\tens\Schur_\Hph\|_{\cb(\Sch^p[E])}.
  \end{gather*}

\end{rem}

This approximation is two-sided whereas the proof of Lemma~\ref{trprp}
uses only a one-sided approximation.  This subtlety makes a difference
if one tries to give a direct proof by complex interpolation, as we
shall do now.

\begin{prp}
  Let $\Ga$~be a discrete amenable group and let $(\mu_\iota)$~be a
  \emph{Reiter net of means} for~$\Ga$:
  \begin{itemize}
  \item each~$\mu_\iota$ is a positive sequence summing to~1 with
    finite support~$\Ga_\iota\subseteq\Ga$ and viewed as a
    diagonal operator from~$\ell^2_{\Ga_\iota}$ to~$\ell^2_\Ga$,
    so that
    \begin{equation*}
      \|\mu_\iota\|_{\Sch^1}=\sum_{\ga\in\Ga_\iota}{(\mu_\iota)}_{\ga}=1;
    \end{equation*}
  \item the net~$(\mu_\iota)$ satisfies, for each~$\ga\in\Ga$,
    \emph{Reiter's Property~$P_1$}:
    \begin{equation}\label{reiter}
      \sum_{\beta\in\Ga}{\bigl|{(\mu_\iota)}_{\ga^{-1}\beta}-{(\mu_\iota)}_{\beta}\bigr|}\to0.
    \end{equation}
  \end{itemize}
  Let~$x\in\mat_n\otimes\vn=\vn(\tr\otimes\tau)$ 
  and~$p\ge1$. Then
  \begin{equation*}
    \limsup\|x\mu_\iota^{1/p}\|_{\Sch^p(\Sch^p_n)}=\|x\|_{\Ell^p(\tr\otimes\tau)}.
  \end{equation*}
\end{prp}

\begin{proof}
  Consider~$x=\sum_{\ga\in\Ga}x_\ga\tens\la_\ga$
  with only a finite number of the~$x_\ga\in\mat_n$ nonzero. As
  \begin{equation*}
    \sum_{\beta\in\Ga}{\bigl|{(\mu_\iota)}_{\ga^{-1}\beta}^{1/2}-{(\mu_\iota)}_{\beta}^{1/2}\bigr|}^2\le
    \sum_{\beta\in\Ga}{\bigl|{(\mu_\iota)}_{\ga^{-1}\beta}-{(\mu_\iota)}_{\beta}\bigr|},
  \end{equation*}
  Property~$P_1$ implies \emph{Property~$P_2$}:
  \begin{equation*}
    \|\la_\ga\mu_\iota^{1/2}-\mu_\iota^{1/2}\la_\ga\|_{\Sch^2}\to0, 
  \end{equation*}
  so that
  \begin{equation*}
    \|x\mu_\iota^{1/2}-\mu_\iota^{1/2}x\|_{\Sch^2(\Sch^2_n)}\to0.
  \end{equation*}
  As the $\mat_n$-valued matrix of~$x$ for the canonical basis of~$\ell^2_\Ga$ is~$(x_{rc^{-1}})_{(r,c)\in\Ga\times\Ga}$,
  \begin{align*}
    \|x\mu_\iota^{1/2}\|_{\Sch^2(\Sch^2_n)}^2
    &=\sum_{(r,c)\in\Ga\times\Ga}\|x_{rc^{-1}}\|_{\Sch^n_2}^2{(\mu_\iota)}_c\\
    &=\sum_{c\in\Ga}{(\mu_\iota)}_c\sum_{r\in\Ga}\|x_{rc^{-1}}\|_{\Sch^n_2}^2\\
    &=\sum_{c\in\Ga}{(\mu_\iota)}_c\|x\|_{\Ell^2(\tr\otimes\tau)}^2=\|x\|_{\Ell^2(\tr\otimes\tau)}^2.
  \end{align*}
By density and continuity, the result extends to all~$x\in \Ell^2(\tr\otimes\tau)$.

  Let us prove now that for~$x\in\Ell^\infty(\tr\otimes\tau)$
  \begin{equation*}
    \limsup\|x\mu_\iota\|_{\Sch^1(\Sch^1_n)}\le\|x\|_{\Ell^1(\tr\otimes\tau)}.
  \end{equation*}
  The polar decomposition~$x=u|x|$ yields a factorisation~$x=ab$
  with~$a=u\abs{x}^{1/2}$ and~$b=\abs{x}^{1/2}$
  in~$\Ell^\infty(\tr\otimes\tau)$ such that
  \begin{gather*}
    \|a\|_{\Ell^2(\tr\otimes\tau)}=\|b\|_{\Ell^2(\tr\otimes\tau)}=\|x\|_{\Ell^1(\tr\otimes\tau)}^{1/2} \\
    \|a\|_{\Ell^\infty(\tr\otimes\tau)}=\|x\|_{\Ell^\infty(\tr\otimes\tau)}^{1/2}.
  \end{gather*}
  Then
  \begin{math}
    x\mu_\iota=a(b\mu_\iota^{1/2}-\mu_\iota^{1/2}b)\mu_\iota^{1/2}+a\mu_\iota^{1/2}b\mu_\iota^{1/2},
  \end{math}
  so that the Cauchy-Schwarz inequality yields
  \begin{align*}
    \|x\mu_\iota\|_{\Sch^1(\Sch^1_n)}
      &\le\|a\|_{\Ell^\infty(\tr\otimes\tau)}
      \|(b\mu_\iota^{1/2}-\mu_\iota^{1/2}b)\mu_\iota^{1/2}\|_{\Sch^1(\Sch^1_n)}
      +\|a\mu_\iota^{1/2}b\mu_\iota^{1/2}\|_{\Sch^1(\Sch^1_n)}\\
      &\le\|a\|_{\Ell^\infty(\tr\otimes\tau)}
      \|b\mu_\iota^{1/2}-\mu_\iota^{1/2}b\|_{\Sch^2(\Sch^2_n)}
      +\|a\|_{\Ell^2(\tr\otimes\tau)}\|b\|_{\Ell^2(\tr\otimes\tau)}
  \end{align*}
  and therefore our claim. Now complex interpolation yields
  \begin{equation}
    \label{ci}
    \limsup\|x\mu_\iota^{1/p}\|_{\Sch^p(\Sch^p_n)}\le\|x\|_{\Ell^p(\tr\otimes\tau)}
  \end{equation}
  for~$x\in\Ell^\infty(\tr\otimes\tau)$ and~$p\in[1,\infty]$.  Indeed,
  let $u$~be the unitary appearing in the polar decomposition of~$x$.
  Consider the function~$f(z)=u\abs{x}^{pz}\mu_\iota^z$ analytic in
  the strip~$0<\Im z<1$ and continuous on its closure: then $f(\iu
  t)$~is a product of unitaries for~$t\in\R$, so that
  \begin{equation*}
    \|f(\iu t)\|_{\vn(\tr\otimes\tau)}=1.
  \end{equation*}
  Also
  \begin{align*}
    \|f(1+\iu t)\|_{\Sch^1(\Sch^1_n)}
    &=\|\abs{x}^p\mu_\iota\|_{\Sch^1(\Sch^1_n)}
  \end{align*}
  As $\Sch^p(\Sch^p_n)$~is the complex interpolation space~$(\Sch^\infty(\mat_n),\Sch^1(\Sch^1_n))_{1/p}$,
  \begin{align*}
    \|x\mu_\iota^{1/p}\|_{\Sch^p(\Sch^p_n)}=
    \|f(1/p)\|_{\Sch^p(\Sch^p_n)} \le
    \||x|^p\mu_\iota\|_{\Sch^1(\Sch^1_n)}^{1/p}.
  \end{align*}
Then, taking the upper limit and using the estimate on~$\Sch^1(\Sch^1_n)$
 \begin{align*}
\limsup \|x\mu_\iota^{1/p}\|_{\Sch^p(\Sch^p_n)}&\le 
\limsup \||x|^p\mu_\iota\|_{\Sch^1(\Sch^1_n)}^{1/p}\\ &\le
\||x|^p\|_{\Ell^1(\tr\otimes\tau)}^{1/p}=\|x\|_{\Ell^p(\tr\otimes\tau)}.
\end{align*}
  The reverse inequality is obtained by duality; first note that for~$y\in{\vn(\tr\otimes\tau)}$, 
\begin{equation*}\lim \tr y\mu_i= \tau (y).\end{equation*}
With the above notation and the inequality for~$p'$,
\begin{align*}
 \|x\|_{\Ell^p}^p=\tau (|x|^p)=\lim \tr |x|^p\mu_i&= \lim
 \tr \mu_i^{1-1/p}|x|^{1-p}u^*x\mu_i^{1/p}\\&\le
\limsup \|\mu_i^{1-1/p}|x|^{1-p}\|_{\Sch^{p'}(\Sch^{p'}_n)}
\|x\mu_i^{1/p}\|_{\Sch^p(\Sch^p_n)}\\&=
\limsup \||x|^{1-p}\mu_i^{1-1/p}\|_{\Sch^{p'}(\Sch^{p'}_n)}
\|x\mu_i^{1/p}\|_{\Sch^p(\Sch^p_n)} \\&\le
\||x|^{1-p}\|_{\Ell^{p'}}\limsup\|x\mu_i^{1/p}\|_{\Sch^p(\Sch^p_n)}
\end{align*}
 so that
  \begin{equation*}
    \limsup\|x\mu_\iota^{1/p}\|_{\Sch^p(\Sch^p_n)}=\|x\|_{\Ell^p(\tr\otimes\tau)}^p.\qedhere
  \end{equation*}
\end{proof}
\begin{rem}Let $\mu$~be any positive diagonal operator with $\tr
  \mu=1$ and $p\ge 2$: then $\|x\mu^{1/p}\|_{\Sch^p(\Sch^p_n)}\le \|x\|_{\Ell^p}$
  for all $x\in \vn(\tr\otimes\tau)$. The Reiter condition is
  only necessary to go below exponent~2.
\end{rem}

In the same way, using interpolation, we can come back to
approximation on both sides using Reiter means, that is
\begin{equation*}\|\mu_\iota^{1/2p}x\mu_\iota^{1/2p}\|_{\Sch^p(\Sch^p_n)}\le \|x\|_{\Ell^p}\end{equation*}
and we have
\begin{equation*}
  \limsup\|\mu_\iota^{1/2p}x\mu_\iota^{1/2p}\|_{\Sch^p(\Sch^p_n)}
  =\|x\|_{\Ell^p(\tr\otimes\tau)}.
\end{equation*} 
Note that this formula is in the same spirit as the first approach of
this section.

\section{Transfer of lacunary sets into lacunary matrix patterns}
\label{sec:lac}



As a first application of Theorem~\ref{trans2}, let us mention that it
provides a shortcut for some arguments in \cite{ha98} as it permits to
transfer lacunary subsets of discrete group $\Ga$ into lacunary matrix
patterns in $\Ga\times\Ga$.  Let us first introduce the following
terminology.

\begin{dfn}\label{unclp:def}
  Let $\Ga$~be a discrete group and~$\La\subseteq\Ga$.  Let $X$~be the
  reduced $\Cont^*$-algebra $\Cont$ of~$\Ga$ or its noncommutative
  Lebesgue space~$\Ell^p$ for~$p\in\iod[1,\infty]$.
  \begin{enumerate}
  \item The set~$\La$ is \emph{unconditional} in~$X$ if the Fourier
    series of every~$x\in X_\La$ converges unconditionally: there is a
    constant~$D$ such that
    \begin{equation*}
      \biggnorm{\sum_{\ga\in\La'}x_\ga\eps_\ga\la_\ga}_{X}
      \le D\|x\|_{X}
    \end{equation*}
    finite~$\La'\subseteq\La$
    and~$\eps_\ga\in\T$. The minimal constant~$D$ is the
    \emph{unconditional constant of~$\La$ in~$X$}.
  \item If~$X=\Cont$, let~$\tilde X=\Sch^\infty\otimes\Cont$; if~$X=\Ell^p$, let~$\tilde X=\Ell^p(\tr\otimes\tau)$. The set~$\La$ is \emph{completely unconditional} in~$X$ if the
    Fourier series of every~$x\in\tilde X_\La$ converges
    unconditionally: there is a constant~$D$ such
    that
  \begin{equation*}
    \biggnorm{\sum_{\ga\in\La'}x_\ga\otimes\eps_\ga\la_\ga}_{\tilde X}
    \le D\|x\|_{\tilde X}
  \end{equation*}
  for finite~$\La'\subseteq\La$
  and~$\eps_\ga\in\T$. The minimal constant~$D$ is the
  \emph{complete unconditional constant of~$\La$ in~$X$}.
  \end{enumerate}
\end{dfn}

Unconditional sets in~$\Ell^p$ have been introduced as ``$\Lambda(p)$
sets'' in \cite[Definition~1.1]{ha98} for $p>2$: if $\Ga$ is abelian,
they are Walter Rudin's $\Lambda(p)$ sets if $p>2$ and his
$\Lambda(2)$ sets if $p<2$: see \cite{ru60,bo01}.  Asma Harcharras
\cite[Definition~1.5, Comments~1.9]{ha98} termed completely
unconditional sets in~$\Ell^p$ ``$\Lambda(p)_{\cb}$ sets''
if~$p\in\io[2,\infty]$, and ``$\mathrm{K}(p)_{\cb}$ sets''
if~$p\in\iog[1,2]$; her definitions are equivalent to ours by the
noncommutative Khinchin inequality. 

Sets that are unconditional in $\Cont$ have been introduced as
``unconditional Sidon sets'' in \cite{bo81a}. If $\Ga$ is amenable,
Fourier multipliers are automatically c.b.\ on $\Cont_\La$, so that
such sets are automatically completely unconditional in~$\Cont$, and
there are at least three more equivalent definitions for the
counterpart of Sidon sets in an abelian group. If $\Ga$ is
nonamenable, these definitions are not all equivalent anymore and our
notion of completely unconditional sets in~$\Cont$ corresponds to Marek Bo\.zejko's ``c.b.\ Sidon sets.''

\begin{dfn}
  \label{uncsp:def}
  Let $1\le p\le\infty$ and $I\subseteq\Row\times\Col$. 
    \begin{enumerate}
  \item The set~$I$ is \emph{unconditional} in~$\Sch^p$ if the matrix
    representation of every~$x\in\Sch^p_I$ converges
    unconditionally: there is a constant~$D$ such that
    \begin{equation*}
      \Bigl\|\sum_{q\in I'}x_q\eps_q\e_q\Bigr\|_p \le D
      \|x\|_p
    \end{equation*}
    for finite~$I'\subseteq I$
    and~$\eps_q\in\T$. The minimal constant~$D$ is the
    \emph{unconditional constant of $I$ in $\Sch^p$}.
  \item The set $I$ is \emph{completely} unconditional in $\Sch^p$ if
    the matrix representation of every~$x\in\Sch^p_I(\Sch^p)$
    converges unconditionally: there is a constant~$D$ such that
    \begin{equation*}\label{cu}
      \Bigl\|\sum_{q\in I'}x_q\tens\eps_q\e_q\Bigr\|_p \le D
      \|x\|_p
    \end{equation*}
    for finite~$I'\subseteq I$
    and~$\eps_q\in\T$. The minimal constant~$D$ is the
    \emph{complete unconditional constant of $I$ in $\Sch^p$}.
  \end{enumerate}
\end{dfn}

Harcharras \cite[Definitions 4.1~and~4.4,
Remarks~4.6$\,(iv)$]{ha98} termed unconditional and completely
unconditional sets in~$\Sch^p$ ``$\sigma(p)$ sets'' and
``$\sigma(p)_{\cb}$ sets,'' respectively; her definitions are
equivalent by the noncommutative Khinchin inequality.

\begin{prp}\label{trans:set}
  Let $\Ga$~be a discrete group. Let $\La\subseteq\Ga$ and consider
  the associated Toeplitz set
  $\HLa=\{(r,c)\in\Ga\times\Ga:\row\col^{-1}\in \La\}$. Let $p\in\iod[1,\infty]$.
  \begin{enumerate}
  \item If $\Ga$ is amenable, then $\La$ is unconditional in $\Ell^p$
    if $\HLa$ is unconditional in $\Sch^p$.
  \item If $\La$ is completely unconditional in $\Ell^p$, then $\HLa$
    is completely unconditional in $\Sch^p$. The converse holds if
    $\Ga$ is amenable.
  \end{enumerate}
\end{prp}

\begin{proof}
  The first part of $(b)$ follows by the argument of the proof of
  \cite[Proposition~4.7]{ha98}: let us sketch it. Consider the
  isometric embedding of the space $\Sch^p_{\mkern-3mu\HLa}(\Sch^p)$ in
  $\Ell^p_\La(\tr\otimes\tr\otimes\tau)$ that is given in the proof of
  Lemma~\ref{trans1} and apply the equivalent Definition~1.5 in
  \cite{ha98} of the complete unconditionality of~$\La$: this gives
  the complete unconditionality of~$\HLa$ in the equivalent
  Definition~4.4 in \cite{ha98}.

  Unconditionality in $\Ell^p$ expresses the uniform boundedness of
  relative unimodular Fourier multipliers on $\Ell^p_\La$; complete
  unconditionality expresses their uniform complete boundedness.
  Unconditionality in $\Sch^p$ expresses the uniform boundedness of
  relative unimodular Schur multipliers on $\Sch^p_{\mkern-3mu\HLa}$; complete
  unconditionality expresses their uniform complete boundedness. The
  second part of $(b)$ follows therefore from
  Theorem~\ref{trans2}$\,(b)$ and $(a)$ follows from
  Lemma~\ref{trprp}$\,(a)$.
\end{proof}

\begin{rem}
  This transfer does not pass to the limit $p=\infty$: Nicholas
  Varopoulos (\cite[Theorem~4.2]{va69}, see \cite[\S\,5]{ne06} for a
  reader's guide) proved that unconditional sets in
  $\Sch^\infty$ are finite unions of patterns whose rows or whose
  columns contain at most one element, and this excludes sets of the
  form $\HLa$ for any infinite $\La$.
\end{rem}

\begin{rem}
  See \cite[Remark~11.3]{ne06} for an illustration of
  Proposition~4.3$\,(b)$ in a particular context.
\end{rem}

\begin{rem}
  Let $p$ be an even integer greater or equal to 4. The existence of a
  $\sigma(p)_\cb$ set that is not a $\sigma(q)$ set for any $q>p$
  \cite[Theorem~4.9]{ha98} becomes a direct consequence of Walter
  Rudin's construction \cite[Theorem~4.8]{ru60} of a $\Lambda(p)$ set
  that is not a $\Lambda(q)$ set for any $q>p$, because this set has
  property $\mathrm B(p/2)$ \cite[Definition~2.4]{ha98} and is
  therefore $\Lambda(p)_\cb$ by \cite[Theorem~1.13]{ha98} (in fact, it
  is even ``1-unconditional'' in $\Ell^p$ because $\mathrm B(p/2)$ is
  ``$p/2$-independence'' \cite[\S\,11]{ne06}.)
\end{rem}

\begin{rem}
  In the same way, Theorem~5.2 in \cite{ha98} becomes a mere
  reformulation of \cite[Proposition 3.6]{ha98} if one remembers that
  the Toeplitz Schur multipliers are 1-complemented in the Schur
  multipliers for an amenable discrete group and for all classical
  norms.  Basically results on $\Lambda(p)_{\cb}$ sets produce results
  on $\sigma(p)_{\cb}$ sets.
\end{rem}

Let us now estimate the complete unconditional constant of sumsets. In
the case~$\Ga=\Z$, Harcharras \cite[Prop.~2.8]{ha98} proved that
a completely unconditional set in~$\Ell^p$ cannot contain the sumset
of characters~$A+A$ for arbitrary large finite sets~$A$: in
particular, if $\La\supseteq A+A$ with $A$ infinite, then $\La$~is not
a completely unconditional set. Her proof provided thus examples
of~$\Lambda(p)$ sets that are not $\Lambda(p)_\cb$ sets.

We generalise Harcharras' result in two directions. Compare~\cite[\S\,1.4]{lr75}.

\begin{prp}\label{har:lp}
  Let $\Ga$~be a discrete group and~$p\ne2$. A completely
  unconditional set in~$\Ell^p$ cannot contain the sumset of two
  arbitrarily large sets. More precisely, let $\Row$~and~$\Col$ be
  subsets of~$\Ga$ with~$\card{\Row}\ge n$ and $\card{\Col}\ge n^3$. Then, for
  any~$p\ge1$, the complete unconditional constant of the
  sumset~$\Row\Col$ in~$\Ell^p$ is at least~$n^{|1/2-1/p|}$.
\end{prp}

\begin{proof} Let $r_1,\dots,r_{n}$ be pairwise
  distinct elements in $\Row$. We shall select inducively elements
  $c_1,\dots,c_{n}$ in $\Col$ such that the
  $r_ic_j$ are pairwise distinct.
Assume there are $c_1,\dots,c_{m-1}$ such that the induction hypothesis
\begin{equation*}
  \forall\, i,k\le n\ \forall\,j,l\le m-1\quad
  (i,j)\ne(k,l)\ \imp\ \row_i\col_j\ne\row_k\col_l.
\end{equation*}
holds. We are looking for an element $c_m\in\Col$ such that 
\begin{equation*}
    \forall\, i,k\le n\ 
    \forall\, l\le m-1\quad 
    \row_i\col_m\ne\row_k\col_l.
  \end{equation*}
  Such an element exists as long as $m\le n$ because the set $\{r_i^{-1}r_kc_l:i,k\le n,\,\allowbreak l\le m-1\}$ has at most $\bigl(n(n-1)+1\bigr)(m-1)<n^3$ elements.  

  The end of the proof is the same as Harcharras'.  The unconditional
  constant of the canonical basis of elementary matrices in~$\Sch^p_n$
  is~$n^{|1/2-1/p|}$; in particular, there is an unimodular Schur
  multiplier~$\bph$ on~$\Sch^p_n$ of norm~$n^{|1/2-1/p|}$ (which is
  also its complete norm, by the way): see \cite[Lemma~8.1.5]{pi98}.
  Let $\La$ be the sumset $\{r_ic_j:i,j\le n\}$;
  as the $r_ic_j$ are pairwise distinct, we may define a
  sequence~$\ph\in\C^\La$ by~$\ph_{\row_i\col_j}=\bph_{i,j}$. By
  Remark~\ref{transg}, the complete norm of the Fourier
  multiplier~$\ph$ on~$\Ell^p_\La$ is bounded below by the complete
  norm of the Schur multiplier~$\bph$ on~$\Sch^p_I$.
\end{proof}
\begin{exa}
  $\La=\{2^i-2^j:i>j\}$~is not a complete~$\Lambda(p)$ set for any~$p\ne2$.  Indeed, \ $\{2^i-2^j\}=\La\cup-\La$ does not and if
  $\La$ did, then also $-\La$ and~$\La\cup-\La$.
\end{exa}
  
\section{\texorpdfstring{Toeplitz Schur multipliers on~$\Sch^p$ for~$p<1$}{Toeplitz Schur multipliers on Sp for p<1}}\label{sec:tsp1}

When~$0<p<1$, a complete characterisation of bounded Schur multiplier of 
Toeplitz type has been obtained by Alexey Alexandrov and Vladimir Peller in \cite[Theorem~5.1]{AlPe02}.
This result was an easy consequence of their deep results on Hankel Schur 
multipliers. The transfer approach provides a direct proof.

\begin{cor}\label{AP}
  Let~$0<p<1$. Let $\Ga$~be a discrete abelian group with dual
  group~$G$. Let $\ph$~be a sequence indexed by~$\Ga$ and define the
  associated Toeplitz matrix~$\Hph\in\C^\HLa$
  by~$\Hph(r,c)=\ph(\row\col^{-1})$ for~$(\row,\col)\in\Ga\times\Ga$.
  Then the following are equivalent:
\begin{enumerate}
\item The sequence~$\ph$ is the Fourier transform of an atomic measure~$\mu=\sum a_g \delta_{g}$ on~$G$ with~$\sum\abs{a_g}^p\le 1$;
\item The Fourier multiplier~$\ph$ is contractive on~$\Ell^p$;
\item The Fourier multiplier~$\ph$ is contractive on~$\Ell^p(\Sch^p)$;
\item The Schur multiplier~$\Hph$ is contractive on~$\Sch^p$;
\item The Schur multiplier~$\Hph$ is contractive on~$\Sch^p(\Sch^p)$.
\end{enumerate}
 \end{cor}

\begin{proof}
  The implication~$(d)\Rightarrow(b)$ follows from
  Lemma~\ref{trprp}$\,(a)$. The equivalence~$(c)\Leftrightarrow(e)$
  follows from Theorem~\ref{trans2}$\,(a)$. The
  characterisation~$(a)\Leftrightarrow(b)$ is an old result of Daniel Oberlin
  \cite{Ober76}. It is plain that $(e)\Rightarrow(d)$.  At last, \
  $(a)\Rightarrow(c)$~is obvious by the~$p\mkern1mu$-triangular
  inequality.
\end{proof}

\begin{rem}\label{b-cb}
  As a consequence, we get that the norm of a Toeplitz Schur
  multiplier on~$\Sch^p(\Sch^p)$ coincides with its norm on~$\Sch^p$
  when~$p<1$. If $p\in\{1,2,\infty\}$, this holds for every Schur
  multiplier. Let $p\in\io[1,2]\cup\io[2,\infty]$. Then we still do
  not know whether Schur multipliers are automatically c.b.\
  on~$\Sch^p$. But from \cite[Proposition~8.1.3]{pi98}, we know that
  $(b)$~and~$(c)$ are not equivalent: if~$\Ga$ is an infinite abelian
  group, there is a bounded Fourier multiplier on~$\Ell^p$ that is not
  c.b.  This example is easy to describe: if an infinite set
  $A\subseteq\Ga$ is lacunary enough, the sumset $A+A$ is
  unconditional in~$\Ell^p$ (see \cite[Theorem~5.13]{lr75}); by
  Proposition~\ref{har:lp}, it cannot be completely unconditional.  In
  particular, this shows that in Lemma~\ref{trprp}$\,(a)$ we cannot
  remove the restriction to truncated Toeplitz matrices in the
  computation of the Schur multiplier norm, that is,
  $(b)\Rightarrow(d)$ does not hold.
\end{rem}

\begin{rem}
  Our questions may also be addressed in the case of a compact group:
  a measurable function~$\ph$ on~$\T$ defines
  \begin{itemize}
  \item the Fourier multiplier on measurable functions on~$\T$ by
    $x\mapsto\ph x$;
  \item the Schur multiplier on integral operators on $\Ell^2(\T)$ with kernel a
    measurable function~$x$ on $\T\times\T$ by $x\mapsto\Hph x$, where
    $\Hph(z,w)=\ph(zw^{-1})$.
  \end{itemize}
  Victor Olevskii \cite{ol96} constructed a continuous function~$\ph$
  that defines a bounded Fourier multiplier on the space of functions
  with $p$-summable Fourier series endowed with the norm given by
  $\norm{x}=\bigl(\sum\abs{\hat x(n)}^p\bigr)^{1/p}$ for every
  $p\in\io[1,\infty]$, while the corresponding Schur
  multiplier is not bounded on the Schatten-von-Neumann class $\Sch^p$
  of operators on $\Ell^2(\T)$ for any $p\in\io[1,2]\cup\io[2,\infty]$.

\end{rem}





\section{{The Riesz projection and the Hilbert
    transform}}\label{sec:rpht}

In this section, we concentrate on~$\Ga=\Z$, the dual group of~$\T$.

\begin{prp}
  \label{rhcb}
  Let $\rh$~be a linear combination of the identity and the upper
  triangular projection of~$\N\times\N$: there are~$z,w\in\C$ so that
  $\rh_{i,j}=z$ if~$i\le j$ and~$\rh_{i,j}=w$ if~$i>j$.  Then the norm
  of the Schur multiplier~$\rh$ on~$\Sch^\psi$ coincides with the norm
  of the Schur multiplier~$\rh$ on $\Sch^\psi(\Sch^\psi)$.
\end{prp}

\begin{proof}
  Let $a\in\Sch^\psi_m(\Sch^\psi_n)$: \ $a$~may be considered as an~$m\times
  m$ matrix~$(a_{ij})$ whose entries~$a_{ij}$ are $n\times n$
  matrices, and $a$~may be identified with the block matrix
  \begin{equation*}
    \tilde{a}=
    \begin{pmatrix}
      0&a_{11}&0&a_{12}&\cdots\\
      0&0&0&0&\cdots\\
      0&a_{21}&0&a_{22}&\cdots\\
      0&0&0&0&\cdots\\
      \vdots&\vdots&\vdots&\vdots&\ddots
    \end{pmatrix}
  \end{equation*}
In this identification, \ $\Id_{\Sch^\psi_n}\otimes\Schur_\rh(a)$ is~$\Schur_\rh(\tilde{a})$.
\end{proof}

The Hilbert transform~$\Ht$ is the Schur multiplier obtained by
choosing $z=-1$ and $w=1$. The upper triangular operators in~$\Sch^p$ can be
seen as a noncommutative $\Hardy^p$ space, and $\Ht$ corresponds
exactly to the Hilbert transform in this setting (see
\cite{Ran98,MaWe98}).  Using classical results on $\Hardy^p$ spaces,
all Hilbert transforms are c.b.\ for~$1<p<\infty$
(see \cite{Zs80,Ran98,MaWe98}).

On the torus~$\T$, the classical Hilbert transform~$H$ corresponds to
the Fourier multiplier given by the sign function (with the
convention~$\sgn(0)=1$) and its norm on~$\Ell^p$ is
\begin{math}
  \cot \left(\pi/2\max(p,p')\right)=\csc(\pi/p)+\cot(\pi/p)
\end{math}
for~$1<p<\infty$. The story of the computation of this norm starts
with a paper by Israel Gohberg and Naum Krupnik \cite{Gk68} for $p$ a
power of~2.  The remaining cases were handled by Stylianos Pichorides
\cite{Pi72} and Brian Cole (see \cite{Gam78}) independently.  The most
achieved results are those of Brian Hollenbeck, Nigel Kalton and Igor
Verbitsky \cite{HoKaVe03}, but they rely on complex variable methods
that are not available in the operator-valued case.  When $p$~is a
power of~2 (or its conjugate), a combination of arguments of Gohberg
and Krupnik \cite{GoKr92} with some of L\'aszl\'o Zsid\'o \cite{Zs80} yields the
following result.

\begin{thm}\label{normh}
  Let $p\in\io[1,\infty]$. The norm and the complete norm of the
  Hilbert transform~$\Ht$ on~$\Sch^p$ coincide with the complete norm
  of the Hilbert transform~$H$ on~$\Ell^p$: if $\Hsgn(i,j)=\sgn(i-j)$
  for~$i,j\ge1$,
  \begin{equation*}
    \|\Hsgn\|_{\Schur(\Sch^p)}=\|\Hsgn\|_{\Schur_\cb(\Sch^p)}=\|\sgn\|_{\Fourier_\cb(\Ell^p)}.
  \end{equation*}
  If $p$~is a power of~2, then these norms coincide with the norm
  of~$H$ on~$\Ell^p$:
  \begin{equation*}
    \|\Hsgn\|_{\Schur(\Sch^p)}=\|\Hsgn\|_{\Schur_\cb(\Sch^p)}=\|\sgn\|_{\Fourier_\cb(\Ell^p)}=\|\sgn\|_{\Fourier(\Ell^p)}=\cot({\pi}/{2p}).
  \end{equation*}
\end{thm}
\begin{proof}
  Let $p\ge 2$. The norm of~$H$ on~$\Ell^p$ is~$\cot({\pi}/{2p})$ and
  the three other norms are equal by the transfer theorem~\ref{trans2}
  and the above proposition: we only need to compute the complete norm
  of~$H$. Let $\tilde H=\Id_{\Sch^p}\otimes H$~be the Hilbert
  transform on~$\Ell^p(\tr\otimes\tau)$. We shall use Mischa Cotlar's
  trick to go from~$\Ell^p$ to~$\Ell^{2p}$: the equality
  \begin{math}
    \sgn i\sgn j+1=\sgn(i+j)\*(\sgn i+\sgn j)
  \end{math}
  shows that
\begin{equation}\label{cotlar}
  (\tilde Hf)(\tilde Hg)+fg=\tilde H\bigl((\tilde Hf)g+f(\tilde Hg)\bigr).
\end{equation}
\noindent \emph{First step.} The function~$\sgn$ is not odd because of
its value in~0: this can be fixed in the following way. Let~$\La=2\Z
+1$. The norm of~$\tilde H$ on~$\Ell^p(\tr\otimes\tau)$ is equal to
its norm on~$\Ell^p_\La(\tr\otimes\tau)$. In fact, let
$D$~be defined by~$Df(z)=zf(z^2)$: \ $D$~is a complete isometry
on~$\Ell^p$ with range~$\Ell^p_\La$ that commutes with~$H$.

\noindent \emph{Second step.} Let $S$~be the real subspace
of~$\Ell_\La^p(\tr\otimes\tau)$ consisting of functions with values
in~$\Sch^p$ so that $f(z)$~is selfadjoint for almost all~$z\in \T$.
Let us apply Vern Paulsen's off-diagonal trick \cite[Lemma~8.1]{pa02}
to show that the norm of~$\tilde H$ on~$\Ell^p$ is equal to its norm
on~$S$.  Let~$f\in\Ell^p_\La(\tr\otimes\tau)$: identifying
$\Sch^p_2(\Sch^p)$ with~$\Sch^p$,
\begin{equation*}
  g(z)= 
  \begin{pmatrix}
    0&f(z)\\
    f(z)^* &0
  \end{pmatrix}
\end{equation*}
defines an element of~$S$.  As adjoining is isometric on~$\Sch^p$,
\begin{equation*}
\|g\|_{S}= 2^{1/p}\|f\|_{\Ell^p(\tr\otimes\tau)}.
\end{equation*}
Let us now consider
\begin{equation*}
  \tilde Hg
  =
  \begin{pmatrix}
    0&\tilde Hf\\
    \tilde H(f^*)&0
  \end{pmatrix}.
\end{equation*}
As $0\notin\La$ by
step~1, the equality~$\sgn(-i)=-\sgn i$ holds for~$i\in\La$: this yields that $\tilde H(f^*)=-(\tilde Hf)^*$. Therefore
\begin{equation*}
  \|\tilde Hg\|_S=2^{1/p}\|\tilde Hf\|_{\Ell^p(\tr\otimes\tau)}.
\end{equation*}
 
\noindent \emph{Third step.} Let $u_p$~be the norm of~$\tilde H$ on~$\Ell^p(\tr\otimes\tau)$: then $u_{2p}\le u_p+\sqrt{1+u_p}$. It
suffices to prove this estimate for~$f\in S$, and by approximation we
may suppose that~$f$ is a finite linear combination of terms~$a_i\tens
z^i+a_i^*\tens z^{-i}$ with~$a_i$ finite matrices. Note that $\tilde
Hf=-(\tilde Hf)^*$. Formula~\eqref{cotlar} with~$f=g$ yields, combined
with H\"older's inequality, 
\begin{equation*}
  \|(\tilde Hf)^2\|_{\Ell^p(\tr\otimes\tau)}
  \le\|f^2\|_{\Ell^p(\tr\otimes\tau)}
  +2u_p\|f\|_{\Ell^{2p}(\tr\otimes\tau)}\|\tilde Hf\|_{\Ell^{2p}(\tr\otimes\tau)}.
\end{equation*}

Since $f$~and~$\tilde Hf$ take normal values,
\begin{gather*}
  \|f^2\|_{\Ell^p(\tr\otimes\tau)}=\|f\|_{\Ell^{2p}(\tr\otimes\tau)}^2\\
  \|(\tilde Hf)^2\|_{\Ell^p(\tr\otimes\tau)}=\|\tilde
  Hf\|_{\Ell^{2p}(\tr\otimes\tau)}^2.
\end{gather*}
Therefore, if
$\|f\|_{\Ell^{2p}(\tr\otimes\tau)}=1$, \ $\|\tilde
Hf\|_{\Ell^{2p}(\tr\otimes\tau)}$ must be smaller than the bigger root of~$t^2-2u_pt-1$, that is
\begin{equation*}
  \|\tilde Hf\|_{\Ell^{2p}(\tr\otimes\tau)}^2\le u_p+\sqrt{ u_p^2+1}
\end{equation*}
and~$u_{2p}\le u_p+\sqrt{u_p^2+1}$.

\noindent \emph{Fourth step.} The multiplier~$H$ is an isometry on~$\Ell^2(\tr\otimes\tau)$, so that $u_2=1=\cot(\pi/4)$. As
$\cot(\th/2)=\cot\th+\sqrt{\cot^2\th+1}$ for~$\th\in\io[0,\pi]$, we
conclude by an induction.
\end{proof}

Unfortunately, we cannot deal with other values of~$p>2$ by this method.

The Riesz projection~$\Rt$ is the Schur multiplier obtained by
choosing $z=0$ and $w=1$ in Proposition~\ref{rhcb}: it is the
projection on the upper triangular part. On the torus, the classical
Riesz projection~$T$, that is the projection onto the analytic part,
corresponds to the Fourier multiplier given by the indicator
function~$\un_{\Z^+}$ of nonnegative integers; its norm
on~$\Ell^p$ has been computed by Hollenbeck and Verbitsky
\cite{HoVe00}: it is
\begin{math} \csc(\pi/p)\end{math}. As for the Hilbert transform, we
know that the norm and the complete norm of~$\Rt$ on~$\Sch^p$ are
equal and coincide with the complete norm of~$T$ on~$\Ell^p$; but,
to the best of our knowledge, there is no simple formula
like~\eqref{cotlar} to go from exponent~$p$ to~$2p$. We only obtained
the following computation.

\begin{prp}\label{normt}
  Let $p\in\io[1,\infty]$. The norm and the complete norm of the Riesz projection~$\Rt$ on~$\Sch^p$ coincide with the complete norm of the Riesz projection~$T$ on~$\Ell^p$: if $\Hun(i,j)=\un_{\Z^+}(i-j)$ for~$i,j\ge1$,
  \begin{equation*}
    \|\Hun\|_{\Schur(\Sch^p)}=\|\Hun\|_{\Schur_\cb(\Sch^p)}=\|\un_{\Z^+}\|_{\Fourier_\cb(\Ell^p)}.
  \end{equation*}
  If~$p=4$, then these norms coincide with the norm of~$T$ on~$\Ell^p$:
  \begin{equation*}
    \|\Hun\|_{\Schur(\Sch^4)}=\|\Hun\|_{\Schur_\cb(\Sch^4)}=\|\un_{\Z^+}\|_{\Fourier_\cb(\Ell^4)}=\|\un_{\Z^+}\|_{\Fourier(\Ell^4)}=\sqrt2.
  \end{equation*}
\end{prp}

\begin{proof}
  We shall compute the norm of~$\Rt$ on~$\Sch^4$. Let $x$~be a finite
  upper triangular matrix and let $y$~be a finite strictly lower
  triangular matrix. We have to prove that
\begin{equation*}
  \sqrt 2 \|x+y\|_{\Sch^4}\ge  \|x\|_{\Sch^4}.
\end{equation*}
Let us make the obvious estimates on~$\Sch^2$ and use the fact that
adjoining is isometric:
\begin{equation*}
  \|\Rt (xx^*)\|_{\Sch^2}
  =\|\Rt ((x+y)x^*)\|_{\Sch^2} 
  \le\|x+y\|_{\Sch^4}\|x\|_{\Sch^4}
\end{equation*}
and similarly, 
\begin{equation*}
  \| (\Id-\Rt) (xx^*)\|_{\Sch^2}=\| (\Id -\Rt)
  (x(x+y)^*)\|_{\Sch^2} \le \|x\|_{\Sch^4} \|x+y\|_{\Sch^4}.
\end{equation*}
As $\Rt$~and~$\Id-\Rt$ have orthogonal ranges,
\begin{equation*}
\|x\|_{\Sch^4}^4=\|xx^*\|_{\Sch^2}^2=\| (\Id-\Rt) (xx^*)\|_{\Sch^2}^2+
\| \Rt (xx^*)\|_{\Sch^2}^2\le 2 \|x\|_{\Sch^4}^2 \|x+y\|_{\Sch^4}^2.\qedhere
\end{equation*}
\end{proof}

\section{Unconditional approximating sequences}\label{sec:uap}




The following definition makes sense for general operator spaces, but
we chose to state it only in our specific context.

\begin{dfn}\label{uap:def}
  Let $\Ga$~be a discrete group and~$\La\subseteq\Ga$. Let $X$~be the
  reduced $\Cont^*$-algebra of~$\Ga$ or its noncommutative Lebesgue
  space~$\Ell^p$ for~$p\in\iod[1,\infty]$.
  \begin{enumerate}
  \item A sequence~$(T_k)$ of operators on~$X_\La$ is an
    \emph{approximating sequence} if each~$T_k$ has finite rank and~$T_kx\to x$ for every~$x\in X_\La$. It is a \emph{complete}
    approximating sequence if the~$T_k$ are uniformly c.b. If $X$
    admits a complete approximating sequence, then $X_\La$ enjoys
    the \emph{c.b.\ approximation property}.
  \item The \emph{difference sequence}~$(\Delta T_k)$ of a sequence~$(T_k)$ is given by~$\Delta T_1=T_1$ and~$\Delta
    T_k=T_k-T_{k-1}$ for~$k\ge2$. An approximating sequence~$(T_k)$ is \emph{unconditional} if the operators
    \begin{equation}
      \label{cuap:eq}
      \sum_{k=1}^n\eps_k\Delta T_k\quad\text{with~$n\ge1$ and~$\eps_k\in\{-1,1\}$}
    \end{equation}
    are uniformly bounded on~$X_\La$: then~$X_\La$ enjoys the
    \emph{unconditional approximation property}.
  \item An approximating sequence~$(T_k)$ is \emph{completely}
    unconditional if the operators in~\eqref{cuap:eq} are uniformly
    c.b.\ on~$X_\La$: then $X_\La$ enjoys the
    \emph{complete} unconditional approximation property. The minimal
    uniform bound of these operators is the \emph{complete
      unconditional constant} of~$X_\La$.
  \end{enumerate}
\end{dfn}


We may always suppose that a complete approximating sequence
on~$\Cont_\La$ is a Fourier multiplier sequence: see
\cite[Theorem~2.1]{hk94}. We may also do so on~$\Ell^p_\La$ if
$\Ell^\infty$ has the so-called QWEP: see \cite[Theorem~4.4]{jr03}.
More precisely, the following proposition holds.

\begin{prp}
  \label{fm}
  Let $\Ga$ be a discrete group and $\La\subseteq\Ga$. Let $X$~either
  be its reduced $\Cont^*$-algebra or its noncommutative Lebesgue
  space~$\Ell^p$, where $p\in\iod[1,\infty]$ and $\Ell^\infty$ has the
  QWEP. If $X_\La$ enjoys the completely unconditional approximation
  property with constant~$D$, then, for every~$D'>D$, there is a
  complete approximating sequence of Fourier multipliers~$(\ph_k)$
  that realises the completely unconditional approximation property
  with constant~$D'$: the Fourier
  multipliers~$\sum_{k=1}^n\eps_k\Delta\ph_k$ are uniformly completely
  bounded by~$D'$ on~$X_\La$.
\end{prp}

Let us now describe how to skip blocks in an approximating sequence in
order to construct an operator that acts like the Riesz projection on
the sumset of two infinite sets. The following trick will be used in
the induction below: compare~\cite[proof of Theorem~4.2]{nptv}:
\begin{equation*}
  \begin{pmatrix}
    1&1\hskip\arraycolsep\vrule\hskip\arraycolsep0\\0&1\hskip\arraycolsep\vrule\hskip\arraycolsep0\\\noalign{\hrule}0&0\hskip\arraycolsep\vrule\hskip\arraycolsep0
  \end{pmatrix}
  -
  \begin{pmatrix}
    1&1\hskip\arraycolsep\vrule\hskip\arraycolsep0\\1&1\hskip\arraycolsep\vrule\hskip\arraycolsep0\\1&1\hskip\arraycolsep\vrule\hskip\arraycolsep0
  \end{pmatrix}
  +
  \begin{pmatrix}
    1&1&1\\1&1&1\\1&1&1
  \end{pmatrix}
  =
  \begin{pmatrix}
    1&1&1\\0&1&1\\0&0&1
  \end{pmatrix}
\end{equation*}
\begin{lem}
  \label{diag}
  Let $\Ga$~be a discrete group and~$\La\subseteq\Ga$.  Suppose that
  $\La$ contains the sumset~$\Row\Col$ of two infinite sets
  $\Row$~and~$\Col$. Let $(T_k)$~be either an approximating sequence
  on~$\Ell^p_\La$ with $p\in\iod[1,\infty]$, or an approximating
  sequence of Fourier multipliers on~$\Cont_\La$. Let~$\eps>0$. There
  is a sequence~$(\row_i)$ in~$\Row$, a sequence~$(\col_i)$ in~$\Col$
  and there are indices~$l_1<k_2<l_2<k_3<\dots$ such that, for
  every~$n$, the \emph{skipped block sum}
  \begin{equation}
    \label{sbs}
    U_n=T_{l_1}+(T_{l_2}-T_{k_2})+\dots+(T_{l_n}-T_{k_n})
  \end{equation}
  acts, up to~$\eps$, as the Riesz projection on the sumset~$\{\row_i\col_j\}_{i,j\le n}$:
  \begin{equation}
    \label{diag1}
    \begin{cases}
      \norm{U_n(\la_{\row_i\col_j})-\la_{\row_i\col_j}}<\eps&\text{if $i\le j\le n$},\\
      \norm{U_n(\la_{\row_i\col_j})}<\eps&\text{if $j<i\le n$}.
    \end{cases}
  \end{equation}
\end{lem}
\begin{proof}
  Let us construct the sequences and indices by induction. If~$n=1$,
  let $\row_1$~and~$\col_1$ be arbitrary; there is~$l_1$ such that
  $\norm{T_{l_1}(\la_{\row_1\col_1})-\la_{\row_1\col_1}}<\eps$.  Suppose that $\row_1,\dots,\row_n$, $\col_1,\dots,\col_n$, $l_1,\dots,k_n,l_n$ have been constructed. Let~$\delta>0$ to be chosen later.
  \begin{itemize}
  \item The operator~$U_n$ defined by Equation~\eqref{sbs} has finite
    rank. If it is a Fourier multiplier, one can choose an
    element~$\row_{n+1}\in \Row$ such that
    $U_n(\la_{\row_{n+1}\col_j})=0$ for~$j\le n$. If it acts
    on~$\Ell^p_\La$ with $p\in\iod[1,\infty]$, one can choose an
    element~$\row_{n+1}\in \Row$ such that
    $\norm{U_n(\la_{\row_{n+1}\col_j})}<\delta$ for~$j\le n$ because
    $(\la_\ga)_{\ga\in\Ga}$ is weakly null in $\Ell^p$.
  \item There is~$k_{n+1}>l_n$ such that
    $\norm{T_{k_{n+1}}(\la_\ga)-\la_\ga}<\delta$ for~$\ga\in\{\row_i\col_j:1\le i\le n+1,1\le j\le n\}$.
  \item Again, choose~$\col_{n+1}\in \Col$ such that $\norm{(U_n-T_{k_{n+1}})(\la_{\row_i\col_{n+1}})}<\delta$ for~$i\le n+1$.
  \item Again, choose~$l_{n+1}>k_{n+1}$ such that $\norm{T_{l_{n+1}}(\la_\ga)-\la_\ga}<\delta$ for~$\ga\in\{\row_i\col_j:1\le i,j\le n+1\}$.
  \end{itemize}
  Let~$U_{n+1}=U_n+(T_{l_{n+1}}-T_{k_{n+1}})$. If $i\le n+1$ and~$j\le n$, then
  \begin{equation*}
    \norm{\Delta U_{n+1}(\la_{r_ic_j})}\le\norm{T_{l_{n+1}}(\la_{r_ic_j})-\la_{r_ic_j}}+\norm{\la_{r_ic_j}-T_{k_{n+1}}(\la_{r_ic_j})}<2\delta,
  \end{equation*}
  so that
  \begin{align*}
    \norm{U_{n+1}(\la_{r_ic_j})-\la_{r_ic_j}}<\eps+2\delta&\quad\text{if $i\le j\le n$}\\
    \norm{U_{n+1}(\la_{r_ic_j})}<\eps+2\delta&\quad\text{if $j<i\le n$}\\
    \norm{U_{n+1}(\la_{r_{n+1}c_j})}<3\delta&\quad\text{if $j\le n$.}
  \end{align*}
If $i\le n+1$, then
\begin{equation*}
  \norm{U_{n+1}(\la_{r_ic_{n+1}})-\la_{r_ic_{n+1}}}
  \le\norm{(U_n-T_{k_{n+1}})(\la_{r_ic_{n+1}})}+\norm{T_{l_{n+1}}(\la_{r_ic_{n+1}})-\la_{r_ic_{n+1}}}
  <2\delta.
\end{equation*}
This shows that our choice of~$\row_{n+1},\col_{n+1},k_{n+1}$ and~$l_{n+1}$ is adequate if
$\delta$~is small enough.
\end{proof}

This construction will provide an obstacle to the unconditionality of sumsets.

\begin{thm}
  \label{T}
  Let $\Ga$~be a discrete group and~$\La\subseteq\Ga$.  Suppose that $\La$
  contains the sumset~$\Row\Col$ of two infinite sets $\Row$~and~$\Col$. 
  \begin{enumerate}
  \item Let $1<p<\infty$. The complete unconditional constant of any approximating
    sequence for~$\Ell^p$ is bounded below by the norm of the
    Riesz projection on~$\Sch^p$, and thus by~$\csc\pi/p$.
  \item The spaces $\Ell^1_\La$~and~$\Cont_\La$ do not enjoy the complete
    unconditional approximation property. 
  \item If $\Ga$~is amenable, then the space~$\Cont_\La$ does not enjoy the unconditional
    approximation property.
  \end{enumerate}
\end{thm}

\begin{proof}
  Let $(T_k)$~be an approximating sequence on~$\Ell^p_\La$.
  By Lemma~\ref{diag}, for every~$\eps>0$ and every~$n$, there are
  elements~$r_1,\dots,r_n\in\Row$, $c_1,\dots,c_n\in\Col$ such that
  the Fourier multiplier~$\ph$ given by the indicator function
  of ${\{r_ic_j\}_{i\le j}}$ is near to a skipped block sum~$U_n$
  of~$(T_k)$ in the sense that
  $\|U_n(\la_{r_ic_j})-\ph_{r_ic_j}\la_{r_ic_j}\|<\eps$. But $U_n$~is
  the mean of two operators of the form~\eqref{cuap:eq}: its complete
  norm will provide a lower bound for the complete unconditional
  constant of~$X_\La$. Let us repeat the argument of
  Lemma~\ref{trans1} with~$x\in\Sch^p_n$: as
  \begin{align*}
    \Bignorm{\sum_{i,j=1}^nx_{i,j}\e_{i,j}}_{\Sch^p_n}
      &=\Bignorm{\Bigl(\sum_{i=1}^n\e_{i,i}\otimes\la_{r_i}\Bigr)\Bigl(\sum_{i,j=1}^nx_{i,j}\e_{i,j}\otimes\la_\id\Bigr)\Bigl(\sum_{j=1}^n\e_{j,j}\otimes\la_{c_j}\Bigr)}_{\Ell^p(\tr\otimes\tau)}\\
      &=\Bignorm{\sum_{i=1}^nx_{i,j}\e_{i,j}\otimes\la_{r_ic_j}}_{\Ell^p(\tr\otimes\tau)}
  \end{align*}
  and
  \begin{equation*}
    \Bignorm{\sum_{i=1}^nx_{i,j}\e_{i,j}\otimes(U_n(\la_{r_ic_j})-\ph_{r_ic_j}\la_{r_ic_j})}_{\Ell^p(\tr\otimes\tau)}<n^2\eps\norm{x}_{\Sch^p_n},
  \end{equation*}
  the complete norm of~$U_n$ is nearly bounded below by the norm of
  the Riesz projection on~$\Sch^p_n$:
  \begin{align*}
    \Bignorm{\sum_{i=1}^nx_{i,j}\e_{i,j}\otimes U_n(\la_{r_ic_j})}_{\Ell^p(\tr\otimes\tau)}
    &>\Bignorm{\sum_{i\le j}x_{i,j}\e_{i,j}\otimes\la_{r_ic_j}}_{\Ell^p(\tr\otimes\tau)}-n^2\eps\norm{x}_{\Sch^p_n}\\
    &=\norm{\Rt(x)}_{\Sch^p_n}-n^2\eps\norm{x}_{\Sch^p_n}.
  \end{align*}
  This proves as well~$(a)$ as the first assertion in~$(b)$, because
  the Riesz projection is unbounded on~$\Sch^1$.  Let $(T_k)$~be an
  approximating sequence on~$\Cont_\La$: by Lemma~\ref{fm}, we may
  suppose that $(T_k)$~is a sequence of Fourier multipliers. Thus the
  second assertion in~$(b)$ follows from Lemma~\ref{diag} combined
  with the preceding argument (where $\Sch^p_n$~is replaced
  by~$\mat_n$ and $\Ell^p(\tr\otimes\tau)$ by~$\mat_n\otimes\Cont$)
  and the unboundedness of the Riesz projection
  on~$\Sch^\infty$. For~$(c)$, note that the Fourier multipliers $T_k$
  are automatically c.b.\ on~$\Cont_\La$ if $\Ga$~is amenable
  \cite[Corollary~1.8]{dh85}.
\end{proof}

Theorem~\ref{T}$\,(b)$ has been devised originally to prove that the
Hardy space $\mathrm H^1$, corresponding to the case
$\La=\N\subseteq\Z$ and $p=1$, admits no completely unconditional
basis: see \cite{ri00,ri01}.  Theorem~\ref{T}$\,(c)$ both generalises
the fact that a sumset cannot be a Sidon set (see
\cite[\S\S\,1.4,\,6.6]{lr75} for two proofs and historical remarks, or
\cite[Proposition~IV.7]{lq04}) and Daniel Li's result
\cite[Corollary~13]{li96} that the space~$\Cont_\La$ does not have the
``metric'' unconditional approximation property if $\Ga$~is abelian
and $\La$ contains a sumset.  Li \cite[Theorem~10]{li96} also
constructed a set $\La\subseteq\Z$ such that $\Cont_\Lambda$ has this
property while $\La$ contains the sumset of arbitrarily large sets.
This theorem also provides a new proof that the disc algebra has no
unconditional basis and answers \cite[Question~6.1.6]{ne99}.

\begin{exa}
  Neither the span of products~$\{r_ir_j\}$ of two Rademacher
  functions in the space of continuous functions on~$\{-1,1\}^\infty$ nor the span of
  products~$\{s_is_j\}$ of two Steinhaus functions
  in the space of continuous functions on~${\T}^\infty$ have an unconditional basis.
\end{exa}




\section{Relative Schur multipliers of rank one}\label{sec:rsm1}

Let $\rh$ be an \emph{elementary} Schur multiplier on $\Sch^\infty$, that is,
\[\rh=x\otimes y=(x_ry_c)_{(\row,\col)\in\Row\times\Col}:\] then its
norm is
$\sup_{\row\in\Row}\abs{x_\row}\sup_{\col\in\Col}\abs{y_\col}$. How is
this norm affected if $\rh$ is only partially specified, that is, if
the action of~$\rh$ is restricted to matrices with a given support?

\begin{thm}
  \label{mps}
  Let $I\subseteq\Row\times\Col$ and consider
  $(x_\row)_{\row\in\Row}$~and~$(y_\col)_{\col\in\Col}$. The relative
  Schur multiplier on~$\Sch^\infty_I$ given by $(x_\row
  y_\col)_{(\row,\col)\in I}$ has norm~$\sup_{(\row,\col)\in
    I}\abs{x_\row y_\col}$.
\end{thm}

Note that the norm of the Schur multiplier $(x_\row
y_\col)_{(\row,\col)\in I}$ is bounded by
\begin{math}
  \sup_{\row\in\Row}\abs{x_\row}\sup_{\col\in\Col}\abs{y_\col}
\end{math}
because the matrix~$(x_\row y_\col)_{(\row,\col)\in\Row\times\Col}$
trivially extends~$(x_\row y_\col)_{(\row,\col)\in I}$; the proof
below provides a constructive nontrivial extension of this Schur
multiplier that is a composition of ampliations of the Schur
multiplier in the following lemma.


\begin{lem}
  \label{schur}
  The Schur multiplier
  \begin{math}
    \begin{pmatrix} \overline z&w\\ \overline w&z\end{pmatrix}
  \end{math}
  has norm~$\max(\abs{z},\abs{w})$ on~$\mat_2$.
\end{lem}
\begin{proof}
  This follows from the decomposition
  \begin{equation*}
    \begin{pmatrix} \overline z&w\\ \overline w&z\end{pmatrix}
    =\frac{\abs{z}+\abs{w}}2
    \begin{pmatrix}
      \bar tu\\t\bar u
    \end{pmatrix}
    \otimes
    \begin{pmatrix}
      \overline{tu}&tu
    \end{pmatrix}
    +\frac{\abs{z}-\abs{w}}2
    \begin{pmatrix}
      \bar tu\\-t\bar u
    \end{pmatrix}
    \otimes
    \begin{pmatrix}
      \overline{tu}&-tu
    \end{pmatrix},
  \end{equation*}
where $t,u\in\T$ are chosen so that $z=\abs{z}t^2$ and $w=\abs{w}u^2$.
 \end{proof}

\begin{proof}[Proof of Theorem~\ref{mps}]
  We may suppose that $\Col$~is the finite set~$\{1,\dots,m\}$ and
  that $\Row$~is the finite set~$\{1,\dots,n\}$, that each~$y_\col$ is
  nonzero and that each row in~$\Row$ contains an element of~$I$.  We
  may also suppose that
  $(\abs{x_\row})_{\row\in\Row}$~and~$(\abs{y_\col})_{\col\in\Col}$
  are nonincreasing sequences. For each~$\row\in\Row$ let $c_r$~be the
  least column index of elements of~$I$ in or above row~$\row$: in
  other words,
  \begin{equation*}
    \col_\row=\min_{\row'\le\row}\min\{\col:(\row',\col)\in I\}.
  \end{equation*}
  The sequence~$(\col_\row)_{\row\in\Row}$ is nonincreasing. Let us
  define its inverse $(\row_\col)_{\col\in\Col}$ in the sense that
  $\row_\col\le\row\Leftrightarrow\col_\row\le\col$: for
  each~$\col\in\Col$ let $\row_\col=\min\{\row:\col_\row\le\col\}$.
  Given~$\row$, let $\row'\le\row$~be such that $(\row',\col_\row)\in
  I$: then $\abs{x_\row
    y_{\col_\row}}\le\abs{x_{\row'}y_{\col_\row}}$, so that
  $\sup_{\row\in\Row}\abs{x_\row y_{\col_\row}}\le\sup_{(\row,\col)\in
    I}\abs{x_\row y_\col}$ and the rank~1 Schur multiplier
  \[\rh_0=(x_\row y_{\col_\row})_{(\row,\col)\in\Row\times\Col}\] with pairwise equal columns is
  bounded by~$\sup_{(\row,\col)\in I}\abs{x_\row y_\col}$ on~$\Sch^\infty_n$.
  We will now ``correct''~$\rh_0$ without increasing its norm so as to make
  it an extension of~$(x_\row y_\col)_{(\row,\col)\in I}$. Let
  $\row\in\Row$~and~$\col'\ge\col_\row$: then
    \begin{align*}
      x_\row y_{\col'}
      =x_\row y_{\col_\row}\frac{y_{\col_\row+1}}{y_{\col_\row}}\cdots\frac{y_{\col'}}{y_{\col'-1}}
      &=x_\row y_{\col_\row}\prod_{\col_\row\le\col\le\col'-1}\frac{y_{\col+1}}{y_{\col}}\\
      &=x_\row y_{\col_\row}\prod_{\substack{\row\ge\row_\col\\\col'\ge\col+1}}\frac{y_{\col+1}}{y_{\col}}.
   \end{align*}
    This shows that it suffices to compose the Schur
    multiplier~$\rh_0$ with the $m-1$ rank~2 Schur multipliers with
    block matrix
    \begin{equation*}
      \rh_{\col}=
      \bordermatrix{
        &\scriptstyle1~\cdots~\col&&\scriptstyle\col+1~\cdots~m\cr
        \begin{matrix}\scriptstyle1\\\scriptstyle\vdots\\\scriptstyle\row_\col-1\end{matrix}
        &\displaystyle\overline{\left(\frac{y_{\col+1}}{y_\col}\right)}&\vrule&1\cr\noalign{\kern-1pt}
        &\multispan3\hrulefill\cr\noalign{\kern-1pt}
        \hfil\begin{matrix}\scriptstyle\row_\col\\\scriptstyle\vdots\\\scriptstyle n\end{matrix}
        &1&\vrule&\displaystyle\frac{y_{\col+1}}{y_\col}\cr
        },
    \end{equation*}
    each of which has norm~1 on~$\Sch^\infty_n$ by Lemma~\ref{schur}. 
\end{proof}

\begin{rem}
  We learnt after submitting this article that Timur Oikhberg proved
  independently Theorem~\ref{mps} and gave some applications to it: compare~\cite{oi10}.
\end{rem}
\begin{rem}
  As an illustration, let $\Col=\Row=\{1,...,n\}$ and~$I=\{(r,c):r\ge
  c\}$, and let $a_i$~be an increasing sequence of positive numbers.
  Take $x_r=a_r$~and~$y_c=1/a_c$. Then the relative Schur
  multiplier~$(a_r/a_c)_{r\le c}$ has norm~1. The above proof actually
  constructs the norm~1 extension
  $\left(\min(a_r/a_c,a_c/a_r)\right)_{(r,c)}$. If we
  put~$a_i=\e^{x_i}$, we recover that $(\e^{-|x_r-x_c|})_{(r,c)}$~is
  positive definite, that is, $\abs{\cdot}$~is a conditionally
  negative function on~$\R$.
\end{rem}\medskip



\noindent
2000 \emph{Mathematics subject classification}:
Primary 47B49; Secondary 43A22, 43A46, 46B28.\medskip

\noindent
\emph{Key words and phrases}: Fourier multiplier, Toeplitz Schur
multiplier, lacunary set, unconditional approximation property,
Hilbert transform, Riesz projection.



\clearpage

\chapter{The Sidon constant of sets with three elements}
\label{scis}
% \author{Stefan Neuwirth}
% \date{}

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\begin{abstract}
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\noindent We solve an elementary extremal problem on trigonometric polynomials
and obtain the exact value of the Sidon constant for sets with three
elements $\{n_0,n_1,n_2\}$: it is \[\sec\left(\pi\gcd(n_1-n_0,n_2-n_0)\slash2{\max|n_i-n_j|}\right).\]
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\end{abstract}
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\section{Introduction}
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\noindent Let $\Lambda=\{\la_0,\la_1,\la_2\}$ be a set of three
frequencies and $\rh_0,\rh_1,\rh_2$ three
positive intensities. We solve the following extremal problem: 

\vskip\abovedisplayskip\setbox0=\hbox{$(\dagger)$\quad}\dimen0=\wd0 
\noindent\box0
$\vcenter{
\advance\hsize by -\dimen0
\noindent
To find $\th_0,\th_1,\th_2$ three phases such that,
putting $c_j=\rh_j\e^{\iu\th_j}$, 
the maximum $\max_t|c_0\ei{\la_0t} + c_1\ei{\la_1t} +
c_2\ei{\la_2t}|$ is minimal.}$
\vskip\belowdisplayskip

This enables us to generalise a result of D. J.~Newman. He solved the
following extremal problem for $\Lambda=\{0,1,2\}$:

\vskip\abovedisplayskip\setbox0=\hbox{$(\ddagger)$\quad}\dimen0=\wd0 
\noindent\box0
$\vcenter{
\advance\hsize by -\dimen0
\noindent
To find $f(t)=c_0\ei{\la_0t}
+c_1\ei{\la_1t} +c_2\ei{\la_2t}$ with
$\|f\|_\infty=\max_t|f(t)|\le1$
such that $\|\widehat{f}\|_1=|c_0|+|c_1|+|c_2|$ is maximal.}$
\vskip\belowdisplayskip 

Note that for such an
$f$, $\|\widehat{f}\|_1$ is the Sidon constant of $\Lambda$. Newman's
argument is the following (see \cite[Chapter 3]{sh51}): by the
parallelogram law, 
\begin{eqnarray*}
\max_t|f(t)|^2&=&\max_t|f(t)|^2\vee|f(t+\pi)|^2\\
&\ge&\max_t\bigl(|f(t)|^2+|f(t+\pi)|^2\bigr)/2\\
&=&\max_t\bigl(|c_0+c_1\ei{t}+c_2\ei{2t}|^2+
               |c_0-c_1\ei{t}+c_2\ei{2t}|^2\bigr)/2\\
&=&\max_t|c_0+c_2\e^{\iu2t}|^2+|c_1|^2=
\bigl(|c_0|+|c_2|\bigr)^2+|c_1|^2\\
&\ge&\bigl(|c_0|+|c_1|+|c_2|\bigr)^2/2
\end{eqnarray*}
and equality holds exactly for multiples and translates of $f(t)=1+2\iu\e^{\iu
t}+\e^{\iu2t}$.

Let us describe this paper briefly. We use a real-variable approach:
Problem $(\dagger)$ reduces to studying a function of form 
$$
\Phi(t,\th)=|1+r\e^{\iu\th}\ei{kt}+s\e^{\iu
lt}|^2\text{ for }r,s>0,~k\ne l\in\Z^*
$$
and more precisely $\Phi^*(\th)=\max_t\Phi(t,\th)$. We
obtain the variations of $\Phi^*$: the point is that we find ``by
hand'' a local minimum of $\Phi^*$ and that any two minima of $\Phi^*$
are separated by a maximum of $\Phi^*$, which corresponds to an
extremal point of $\Phi$ and therefore has a handy description. The
solution to Problem $(\ddagger)$ then turns out to derive easily from
this. 

The initial motivation was twofold. In the first place, we wanted to
decide whether sets 
$\Lambda=\{\la_n\}$ such that $\la_{n+1}/\la_n$ is bounded
by some $q$ may have a Sidon constant arbitrarily close to 1 and to
find evidence among sets with three elements. That there are such
sets, arbitrarily large but finite, may in fact 
be proven by the method of Riesz products in \cite[Appendix V,
\S1.II]{ks63}. In the second place, we wished to show that 
the real and complex unconditionality constants are distinct for basic
sequences of characters $\ei{nt}$; we prove however that they
coincide in the space $\mathscr{C}(\T)$ for sequences with three terms. 

\begin{Not}
  $\T=\{\,z\in\C:|z|=1\,\}$ and $\e_\la(z)=z^\la$ for $z\in\T$ and
  $\la\in\Z$.
\end{Not}
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\section{Definitions}
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\begin{dfn}
%
$(1)$ Let $\Lambda\subseteq\Z$. $\Lambda$ is a Sidon set if there is a
constant $C$ such that for all trigonometric polynomials
$f(t)=\sum_{\lambda\in\Lambda}c_\lambda\e^{\iu\lambda t}$ with
spectrum in $\Lambda$ we have 
$$
\|\widehat{f}\|_1=\sum_{\lambda\in\Lambda}|c_\lambda|\le
C\max_t|f(t)|=\|f\|_\infty.
$$
The optimal $C$ is called the Sidon constant of $\Lambda$.

$(2)$ Let $X$ be a Banach space. The sequence $(x_n)\subseteq X$ is a real
\klammer{vs.~complex} unconditional basic sequence in $X$ if there is a
constant $C$ 
such that 
$$\Bigl\|\sum\vartheta_nc_nx_n\Bigr\|_X\le C\Bigl\|\sum c_nx_n\Bigr\|_X$$
for every real \klammer{vs.~complex} choice of signs $\vartheta_n\in\{-1,1\}$
\klammer{vs.~$\vartheta_n\in\T$}  
and every finitely supported family of coefficients $(c_n)$. The
optimal $C$ is the real \klammer{vs.~complex} unconditionality constant of
$(x_n)$ in $X$.
%
\end{dfn}
%
Let us state the two following well known facts.
%
\begin{prp}
%
$(1)$ The Sidon constant of $\Lambda$ is the complex unconditionality
constant of the sequence of functions 
$(\e_\lambda)_{\lambda\in\Lambda}$ in the space $\mathscr{C}(\T)$.

$(2)$ The complex unconditionality constant is at most $\pi/2$ times
the real unconditionality constant.
%
\end{prp}

\begin{proof}
  $(1)$ holds because $\bigl\|\sum\vartheta_\lambda
  c_\lambda\e_\lambda\bigr\|_\infty= \sum|c_\lambda|$ for
  $\vartheta_\lambda=\overline{c_\lambda}\slash|c_\lambda|$.

  $(2)$ Because the complex unconditionality constant of the sequence
  $(\epsilon_n)$ of Rade\-macher functions in
  $\mathscr{C}(\{-1,1\}^\infty)$ is $\pi/2$ (see \cite{se97}),
  \begin{eqnarray*}
    \sup_{\vartheta_n\in\T}\Bigl\|\sum\vartheta_nc_nx_n\Bigr\|_X
    &=&\sup_{x^*\in B_{X^*}}\sup_{\vartheta_n\in\T}\sup_{\epsilon_n=\pm1}
    \Bigl|\sum\vartheta_n c_n\langle x^*,x_n\rangle\epsilon_n\Bigr|\\&\le&
    \pi/2\sup_{x^*\in  B_{X^*}}\sup_{\epsilon_n=\pm1}
    \Bigl|\sum c_n\langle x^*,x_n\rangle\epsilon_n\Bigr|\\
    &=&\pi/2\sup_{\epsilon_n=\pm1}\Bigl\|\sum \epsilon_nc_nx_n\Bigr\|_X.
  \end{eqnarray*}
  Furthermore the real unconditionality constant of $(\epsilon_n)$ in
  $\mathscr{C}(\{-1,1\}^\infty)$ is 1: therefore the factor $\pi/2$
  is optimal.
\end{proof}

Let us straighten out the expression of the Sidon constant. For 
$$f(t)=c_0\ei{\la_0t}+c_1\ei{\la_1t}+
c_2\e^{\iu\la_2t},~c_j=\rh_j\e^{\iu\th_j},$$
the supremum norm $\|f\|_\infty$ of $f$ is equal to 
\begin{equation}
\label{ubc:angle}
\|\rh_0+\rh_1\e^{\iu\th}\e_{\la_1-\la_0}+
\rh_2\e_{\la_2-\la_0}\|_\infty,~
\th=\frac{\la_1-\la_2}{\la_2-\la_0}\th_0
+\th_1+\frac{\la_0-\la_1}{\la_2-\la_0}\th_2
\end{equation}
and therefore the Sidon constant $C$ of
$\Lambda=\{\la_0,\la_1,\la_2\}$ may be written
\begin{equation}
  \label{ubc:ubc}
C=\max_{r,s>0,\th}
(1+r+s)/\|1+r\e^{\iu\th}\e_k+s\e_l\|_\infty\quad\text{with }
\left\{
\begin{array}{l}
k=\la_1-\la_0\\
l=\la_2-\la_0.
\end{array}\right.  
\end{equation}
By change of variables, we may suppose w.l.o.g.~that $k$ and $l$ are
coprime. 
%
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\section{\texorpdfstring{A solution to Extremal problem $(\dagger)$}{A solution to Extremal problem (\textdagger)}}
%
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%
%
\noindent Let us first establish
\begin{lem}\label{ubc:lem}
%
Let $\la_1,\dots,\la_k\in\Z^*$ and $\rh_1,\dots,\rh_k>0$.
Let 
$$
f(t,\th)=1+\rh_1\e^{\iu(\la_1t+\th_1)} + \dots +
\rh_{k-1}\e^{\iu(\la_{k-1}t+\th_{k-1})} +
\rh_k\e^{\iu\la_kt}
$$
and $\Phi(t,\th)=|f(t,\th)|^2$. The critical points $(t,\th)$
such that $\nabla\Phi(t,\th)=0$ satisfy either
$f(t,\th)=0$ or
$\la_1t+\th_1 \equiv \dots \equiv \la_{k-1}t+\th_{k-1} \equiv
\la_kt \equiv0\mod{\pi}$.
\end{lem}
%

\begin{proof}
  
  As $\Phi=(\Re f)^2+(\Im f)^2$, the critical points $(t,\th)$ satisfy
$$
\left\{\vcenter{\halign{\hfil$#$&\,\hfil$#$&$\quad#$\cr \Re\frac{\partial
        f}{\partial t}(t,\th)\,\Re f(t,\th)\,+& \Im\frac{\partial
        f}{\partial t}(t,\th)\,\Im f(t,\th)=0&\cr -\sin(\la_i
      t+\th_i)\,\Re f(t,\th)\,+& \cos(\la_i t+\th_i)\,\Im
      f(t,\th)=0&(1\le i\le k-1),\cr}}\right.
$$
which simplifies to
$$
-\sin(\la_i t+\th_i)\,\Re f(t,\th)\,+ \cos(\la_i t+\th_i)\,\Im
f(t,\th)=0\quad(1\le i\le k,~\th_k=0).
$$
Suppose that $f(t,\th)\ne0$: then the system above implies that
$$
-\sin(\la_i t+\th_i)\cos(\la_j t+\th_j)+\cos(\la_i t+\th_i)\sin(\la_j
t+\th_j)=0~(1\le i,j\le k,~\th_k=0)
$$
and it simplifies therefore to
\begin{equation*}
  \sin(\la_i t+\th_i)=0\quad(1\le i\le k,~\th_k=0).\qedhere
\end{equation*}
\end{proof}

The following result is the core of the paper.
%
\begin{lem}\label{ubc:prp}
%
Let $r,s>0$, $k,l\in\Z^*$ distinct and coprime. Let
\begin{eqnarray*}
\Phi(t,\th)&=&|1+r\e^{\iu\th}\ei{kt}+s\ei{lt}|^2\\
&=&1+r^2+s^2+2r\cos(kt+\th)+2s\cos{lt}+2rs\cos((l-k)t-\th).
\end{eqnarray*}
Let $\Phi^*(\th)=\max_t\Phi(t,\th)$. Then
$\Phi^*$ is an even function with period $2\pi\slash|l|$ that decreases on
$[0,\pi/|l|]$. Therefore $\min_\th\Phi^*(\th)=\Phi^*(\pi/l)$. 
%
\end{lem}

\begin{proof}
  % Let us first locate By Lemma \ref{ubc:lem}, the critical points of
  % $\Phi$ satisfy either $\Phi(t,\th)=0$ or . $\nabla\Phi(t,\th)=0$
  % is Equation \eqref{ubs:S} with
%$$(a_{ij})=\pmatrix{
%  rk&sl&rs(l-k)\cr
%  r&0&-rs\cr}\quad\text{and}\quad\cases{\alpha=kt+\th&\cr\beta=lt.\cr}
%$$ 
%In the notation of Lemma \ref{ubc:lem}, $d_1=-rsl\ne0$,
% $d_2=-r^2sl\ne0$, $d_3=-rs^2l\ne0$. The solution \eqref{ubc:sol}, if
% it exists, yields $\Phi(t,\th)=0$ and corresponds to the absolute
% minimum of $\Phi$. Every other critical point of $\Phi$ satisfies
%$$
%  \label{ubc:lem:cons}
%  kt+\th\equiv lt\equiv0\text{ mod.~}\pi.
%$$
%
  $\Phi^*$ is continuous (see \cite[Chapter 5.4]{ps72}) and even, as
  $\Phi(t,-\th)=\Phi(-t,\th)$. $\Phi^*$ is $(2\pi/|l|)$-periodical:
  let $j\in\Z$ be such that $jk\equiv1$ mod.\ $l$. Then
$$\Phi(t+2j\pi/l,\th) = |1+r\e^{\iu(\th+2\pi
  jk/l)}\ei{kt}+s\ei{lt}|^2 = \Phi(t,\th+2\pi/l).$$ Thus
$\Phi^*$ attains its minimum on $[0,\pi/|l|]$. Furthermore, we have
$$\Phi(-t-2j\pi/l,\pi/l-\th) = \Phi(t+2j\pi/l,-\pi/l+\th) =
\Phi(t,\pi/l+\th),$$ so that $\Phi^*$ has an extremum at $\pi/l$. Now
$$\Phi^*(\pi/l+\th)=\Phi^*(\pi/l)+|\th|\max_{\Phi(t,\pi/l)
  = \Phi^*(\pi/l)}\Bigl|\frac{\partial\Phi}{\partial\th}(t,\pi/l)\Bigr|
+ o(\th).$$ Choose a $t$ such that $\Phi(t,\pi/l)=\Phi^*(\pi/l)$. If
$\partial\Phi/\partial\th(t,\pi/l)\ne0$, then this shows that $\Phi^*$
has a local minimum and a cusp at $\pi/l$. Let us now suppose that
$\partial\Phi/\partial\th(t,\pi/l)=0$. If $\Phi^*$ had a local maximum
at $\pi/l$, then $(t,\pi/l)$ would be a critical point of $\Phi$, so
that by Lemma \ref{ubc:lem} $\cos(kt+\pi/l)=\delta$,
$\cos{lt}=\epsilon$, $\cos((l-k)t-\pi/l)=\delta\epsilon$ for some
$\delta,\epsilon\in\{-1,1\}$. One necessarily would have
$(\delta,\epsilon)\ne(1,1)$. Furthermore,
\begin{eqnarray*}
  \frac{\partial^2\Phi}{\partial\th^2}(t,\pi/l)&=&-2r\delta(1+s\epsilon)\le0\\
  \left|\begin{matrix}{\partial^2\Phi/\partial t^2}&
      {\partial^2\Phi/\partial t\partial\th}\\
      {\partial^2\Phi/\partial\th\partial t}&
      {\partial^2\Phi/\partial\th^2}
    \end{matrix}\right| (t,\pi/l) &=&
  4rsl^2(\delta\epsilon+r\epsilon+s\delta)\ge0,
\end{eqnarray*}
which would imply $\epsilon=-1$, $r=0$, $s=1$. Therefore $\Phi^*$ has
a local minimum at $\pi/l$.  Let us show that then $\Phi^*$ must
decrease on $[0,\pi/|l|]$. Otherwise there are
$0\le\th_0<\th_1\le\pi/|l|$ such that $\Phi^*(\th_1)>\Phi^*(\th_0)$.
As $\pi/|l|$ is a local minimum, there is a $\th_0<\th^*<\pi/|l|$ such
that
$$\Phi^*(\th^*) =
\max_{\th_0\le\th\le\pi/|l|}\Phi^*(\th) = \max_{\substack{0\le t<2\pi\\
\th_0\le\th\le\pi/|l|}}\Phi(t,\th),$$ i.e., 
there further is some $t^*$ such that $\Phi$ has a local maximum at
$(t^*,\th^*)$. But then $kt^*+\th^*\equiv lt^*\equiv0\mod{\pi}$ and
$\th^*\equiv0\mod{\pi/l}$ and this is false.
\end{proof}

By Computation \eqref{ubc:angle} and Lemma \ref{ubc:prp}, we obtain 
%
\begin{thm}\label{spe1}
Let $\la_0,\la_1,\la_2\in\R$ and
$\rh_0,\rh_1,\rh_2>0$. The solution to Extremal
problem $(\dagger)$ is the following.
\begin{itemize}
\item If the smallest additive group containing
$\la_1-\la_0$ and $\la_2-\la_0$ is dense in $\R$,
then the maximum is independent of the phases
$\th_0,\th_1,\th_2$ and makes
$\rh_0+\rh_1+\rh_2$.  

\item Otherwise let $d=\gcd(\la_1-\la_0,\la_2-\la_0)$
be a generator of this group. Then the sought phases
$\th_0,\th_1,\th_2$ are given by
$$
\th_0(\la_2-\la_1)+
\th_1(\la_0-\la_2)+
\th_2(\la_1-\la_0)
\equiv d\pi\mod{2d\pi}.
$$
In particular, these phases may be chosen among 0 and $\pi$.
\end{itemize}
\end{thm}
%
%
%
%
%
%
%
%
\section{\texorpdfstring{A solution to Extremal problem $(\ddagger)$}{A solution to Extremal problem (\textdaggerdbl)}}
%
%
%
%
%
%
%
\noindent There are two cases where one can make explicit computations by Lemma
\ref{ubc:prp}.  

\begin{exa}
  The real and complex unconditionality
  constant of $\{0,1,2\}$ in $\mathscr{C}(\T)$ is $\sqrt{2}$. Indeed,
  a case study shows that
$$\|1+\iu r\e_1+s\e_2\|_\infty = 
\begin{cases}
  r+|s-1|&\text{if $r|s-1|\ge4s$}\\ 
  (1+s)(1+r^2/4s)^{1/2}&\text{if $r|s-1|\le4s$}
\end{cases}$$ and this permits to compute the maximum
\eqref{ubc:ubc}, which is obtained for $r=2$, $s=1$. This yields
another proof to Newman's result presented in the
Introduction.
\end{exa}

\begin{exa}
  The real and complex unconditionality constant of $\{0,1,3\}$
  in $\mathscr{C}(\T)$ is $2/\sqrt{3}$. Indeed, a case study shows
  that $\|1+r\ei{\pi/3}\e_1+s\e_3\|_\infty$ makes
$$\begin{cases}
  1+r-s&\text{if $s\le r/(4r+9)$}\\
  \bigl(\frac{2}{27}s(r^2+9+3r/s)^{3/2}
  -\frac{2}{27}r^3s+\frac{2}{3}r^2+rs+s^2+1\bigr)^{1/2} &\text{if $s\ge
  r/(4r+9)$}
\end{cases}$$ and this permits to compute the maximum
\eqref{ubc:ubc}, which is obtained exactly at $r=3/2$,
$s=1/2$.
\end{exa}

These examples are particular cases of the following theorem.

\begin{thm}\label{ubc:THM}
Let $\la_0,\la_1,\la_2\in\Z$ be distinct. Then the 
Sidon\index{Sidon set!constant} constant of
$\Lambda=\{\la_0,\la_1,\la_2\}$ is 
$\sec(\pi/2n)$, where
$n=\max|\la_i-\la_j|\slash\gcd(\la_1-\la_0,\la_2-\la_0)$.

%
\end{thm}

\begin{proof}  
  We may suppose $\la_0<\la_1<\la_2$. Let
  $k=(\la_1-\la_0)\slash\gcd(\la_1-\la_0,\la_2-\la_0)$ and
  $l=(\la_2-\la_0)\slash\gcd(\la_1-\la_0,\la_2-\la_0)$.  By Lemma
  \ref{ubc:prp}, the Arithmetic-Geometric Mean Inequality bounds the
  Sidon constant $C$ of $\{0,k,l\}$ in the following way:
  \begin{eqnarray*}
    C=\max_{r,s>0}\frac{1+r+s}{\|1+r\e^{\iu\pi/l}\e_k+s\e_l\|_\infty}&\le&
    \max_{r,s>0}\frac{1+r+s}{|1+r\e^{\iu\pi/l}+s|}\\&=&
    \max_{r,s>0}\biggl(1-\sin^2\frac{\pi}{2l}\frac{4r(1+s)}{(1+r+s)^2}\biggr)^{-1/2}\\
    &\le&
    \bigl(1-\sin^2(\pi/2l)\bigr)^{-1/2}=\sec(\pi/2l).
  \end{eqnarray*}
  This inequality is sharp: we have equality for $s=k\slash(l-k)$ and
  $r=1+s$. In fact the derivative of $|1+r\e^{\iu\pi/l}\e^{\iu
    kt}+s\ei{lt}|^2$ is then
$$
\frac{8kl}{ k-l}\cos\frac{kt+\pi/l}{2}\sin\frac{lt}{2}
\cos\frac{(l-k)t-\pi/l}{2},
$$
so that its critical points are
$$
\frac{2j+1}{ k}\pi-\frac{\pi}{ kl},~ \frac{2j }{ l }\pi,~ \frac{2j+1}{
  l-k}\pi+\frac{\pi}{ l(l-k)}:j\in\Z,$$ where it makes
$$4s^2\sin^2\frac{2j+1+l}{2k    }\pi,~
4r^2\cos^2\frac{2j+1 }{2l }\pi,~ 4
\cos^2\frac{2j+1+k}{2(l-k)}\pi:j\in\Z.$$ Therefore the maximum of
$|1+r\e^{\iu\pi/l}\ei{kt}+s\ei{lt}|$ is $2r\cos(\pi/2l)$.
\end{proof}
 
This proof and \eqref{ubc:angle} yield also the
  more precise 
\begin{prp}\label{spe3}
Let $\Lambda=\{\la_0,\la_1,\la_2\}\subseteq\Z$. The solution to
Extremal problem
$(\ddagger)$ is a multiple of
$$
f(t)=
\epsilon_0\,|\la_1-\la_2|\,\ei{\la_0t}+
\epsilon_1\,|\la_0-\la_2|\,\ei{\la_1t}+
\epsilon_2\,|\la_0-\la_1|\,\ei{\la_2t}
$$
with $\epsilon_0,\epsilon_1,\epsilon_2\in\{-1,1\}$ real signs such
that
\begin{itemize}
\item $\epsilon_0\epsilon_1=-1$ if $2^j\mid \la_1-\la_0$
and $2^j\nmid \la_2-\la_0$ for some $j$;

\item $\epsilon_0\epsilon_2=-1$ if $2^j\nmid \la_1-\la_0$
and $2^j\mid \la_2-\la_0$ for some $j$;

\item $\epsilon_1\epsilon_2=-1$ otherwise.
\end{itemize}
The Sidon constant of $\Lambda$ is attained for this $f$. Therefore
the 
complex and real unconditionality constants of
$\{\e_\la\}_{\la\in\Lambda}$ in $\mathscr{C}(\T)$ 
coincide for sets $\Lambda$  with three elements.
\end{prp}






\section{Some consequences}







\noindent Let us underline the following consequences of our computation. 
\begin{cor}
\begin{itemize}
\item[$(1)$] The Sidon constant of sets with three elements is at
most $\sqrt{2}$.

\item[$(2)$] The Sidon constant of $\{0,n,2n\}$ is $\sqrt{2}$,
while the Sidon constant of $\{0,n+1,2n\}$ is at most
$\sec(\pi/2n)=1+\pi^2/8n^2+o(n^{-2})$ and thus arbitrarily close
to 1.

\item[$(3)$] The Sidon constant of
$\{\la_0<\la_1<\la_2\}$ does not depend on
$\la_1$ but on the g.c.d.~of $\la_1-\la_0$ and
$\la_2-\la_0$.
\end{itemize}
\end{cor}

Theorem \ref{ubc:THM} also shows anew that no set of integers with
more than two elements has Sidon constant 1 (see \cite[p.~21]{sh51} or \cite{chm81}). 
Recall now that
$\Lambda=\{\la_n\}\subseteq\Z$ is a Hadamard set if there is 
a $q>1$ such that $|\la_{n+1}/\la_n|\ge q$ for all $n$.
By \cite[Cor.~9.4]{ne98}, the Sidon constant of
$\Lambda$ is at most $1+\pi^2\slash(2q^2-2-\pi^2)$ if 
$q>\sqrt{\pi^2/2+1}\approx2.44$. On the other hand
Theorem \ref{ubc:THM} shows 
\begin{cor}
\begin{itemize}
\item[$(1)$] If there is an integer $q\ge2$ such
that $\Lambda\supseteq\{\la,\la+\mu,\la+q\mu\}$ for some
integers $\la$ and $\mu$, then the Sidon constant of $\Lambda$ is
at least 
$$\sec(\pi/2q)>1+\pi^2/(8q^2).$$

\item[$(2)$] In particular, we have the following bounds for the Sidon
constant $C$ of the set $\Lambda=\{q^k\}$, $q\in\Z\setminus\{0,\pm1,\pm2\}$:
$$
1+\frac{\pi^2}{8\max(-q,q+1)^2}< C\le 
1+\frac{\pi^2}{2q^2-2-\pi^2}.$$
\end{itemize}
\end{cor}
%
%
%
%
%
%
%
%
\section{Three questions}
%
%
%
%
%
%
%
\begin{enumerate}
\item Is there a set $\Lambda$ for which the real and
    complex unconditionality constants of
    $\{\e_\lambda\}_{\lambda\in\Lambda}$ in $\mathscr{C}(\T)$ differ?
    The same question is open in spaces $L^p(\T)$, $1\le p<\infty$,
    and even for the case of three element sets if $p$ is not a small
    even integer, and especially for the set $\{0,1,2,3\}$ in any
    space but $L^2(\T)$.
  \item Let $q>1$. Are there infinite sets
    $\Lambda=\{\lambda_n\}$ such that $|\lambda_{n+1}\slash\lambda_n|\le q$
    with Sidon constant arbitrarily close to 1? What about the
    sequence of integer parts of the powers of a transcendental number
    $\sigma>1$ (see \cite[Cor.\ 2.10, Prop.\
    3.2]{ne98})?
  \item The only set with more than three elements with known
    Sidon constant is $\{0,1,2,3,4\}$, for which it makes 2 (see
    \cite[Chapter 3]{sh51}). Can one compute the Sidon constant of
    sets with four elements? I conjecture that the Sidon constant of
    $\{0,1,2,3\}$ is $5/3$.
\end{enumerate}
% 
%
%





% \def\cprime{$'$} \def\cprime{$'$}
% \begin{thebibliography}{1}

% \bibitem{chm81}
% D.~I. Cartwright, R.~B. Howlett and J.~R. McMullen, {\em Extreme values for the
%   {S}idon constant}, Proc. Amer. Math. Soc. 81 (1981),  531--537.

% \bibitem{ks63}
% J.-P. Kahane and R.~Salem, {\em Ensembles parfaits et s\'eries
%   trigonom\'etriques}, Hermann, 1963.
% \newblock Actualit\'es Scientifiques et Industrielles 1301.

% \bibitem{ne98}
% S.~Neuwirth, {\em Metric unconditionality and {F}ourier analysis}, Studia Math.
%   131 (1998),  19--62.

% \bibitem{ps72}
% G.~P{\'o}lya and G.~Szeg{\H{o}}, {\em Problems and theorems in analysis, {V}ol.
%   {I}: {S}eries, integral calculus, theory of functions}, Springer, 1972.
% \newblock Grundlehren der mathematischen Wissenschaften 193.

% \bibitem{se97}
% J.~A. Seigner, {\em Rademacher variables in connection with complex scalars},
%   Acta Math. Univ. Comenian. (N.S.) 66 (1997),  329--336.

% \bibitem{sh51}
% H.~S. Shapiro, {\em Extremal problems for polynomials and power series},
%   Master's thesis, Massachusetts Institute of Technology, 1951.

% \end{thebibliography}



% \end{document}


%%% Local Variables: 
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\clearpage

\chapter{The maximum modulus of a trigonometric trinomial}\label{tmmoatt}

\begin{abstract}
  \noindent Let $\La$ be a set of three integers and let $\Cont_\Lambda$ be
  the space of $2\pi$-periodic functions with spectrum in $\La$
  endowed with the maximum modulus norm. We isolate the maximum
  modulus points $x$ of trigonometric trinomials $T\in\Cont_\La$
  and prove that $x$ is unique unless $\abs T$ has an axis of
  symmetry. This enables us to compute the exposed and the extreme points
  of the unit ball of $\Cont_\La$, to describe how the maximum
  modulus of $T$ varies with respect to the arguments of its Fourier
  coefficients and to compute the norm of unimodular relative Fourier
  multipliers on $\Cont_\La$. We obtain in particular the Sidon
  constant of $\La$.
\end{abstract}

\section{Introduction}

\noindent Let $\la_1$, $\la_2$ and $\la_3$ be three pairwise distinct integers.
Let $r_1$, $r_2$ and $r_3$ be three positive real numbers. Given three
real numbers $t_1$, $t_2$ and $t_3$, let us consider the
\emph{trigonometric trinomial}
\begin{equation}
  \label{tt}
  T(x)=r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}
\end{equation}
for $x\in\R$. The $\la$'s are the \emph{frequencies} of the
trigonometric trinomial $T$, the $r$'s are the \emph{moduli} or
\emph{intensities} and the $t$'s the \emph{arguments} or \emph{phases}
of its \emph{Fourier coefficients} $r_1\ei{t_1}$, $r_2\ei{t_2}$ and
$r_3\ei{t_3}$.


\begin{figure}[htbm]
  \centering
  \begin{pspicture}(-2.6875,-3.14805)(3.33333,3.14805)
    \parametricplot[plotstyle=ccurve,linewidth=.4pt]{0}{360}{t cos 2
      mul 3 div t 2 mul cos 8 mul 3 div add t sin 2 mul 3 div t 2 mul
      sin 8 mul 3 div sub}
    \pscircle[linewidth=.4pt](0,0){.66667}
    \psline[linewidth=.7pt,linestyle=dotted,dotsep=1.5pt](-.66667,0)(3.33333,0)
    \psline[linewidth=.4pt,linestyle=dashed,dash=2pt 2pt](-1.08152,
    3.10770)(-.33333,-.57735)(3.23211,.61723)
    \rput(-0.90,0){\small$-1$} \rput(-.65,-.80){\small$-\ei{\pi/3}$}
  \end{pspicture}
  \caption{The unit circle, the hypotrochoid $H$ with equation
    $z=4\mkern1mu\emi{2x}+\ei{x}$, the segment from $-1$ to the unique point on
    $H$ at maximum distance and the segments from $-\ei{\pi/3}$ to
    the two points on $H$ at maximum distance.}
  \label{fig:hypotrochoid}
\end{figure}
The maximum modulus of a trigonometric trinomial has an interpretation
in plane geometry. Without loss of generality, we may assume that
$\la_2$ is between $\la_1$ and $\la_3$.  Let $H$ be the curve with
complex equation
\begin{equation}
  \label{hypo}
z=r_1\ei{(t_1-(\la_2-\la_1)x)}+r_3\ei{(t_3+(\la_3-\la_2)x)}\quad
(-\pi<x\le\pi)\text.
\end{equation}
$H$ is a \emph{hypotrochoid}: it is drawn by a point at distance $r_3$
to the centre of a circle with radius
$r_1\abs{\la_2-\la_1}/\abs{\la_3-\la_2}$ that rolls inside another
circle with radius $r_1\abs{\la_3-\la_1}/\abs{\la_3-\la_2}$. The
maximum modulus of \eqref{tt} is the maximum distance of points $z\in
H$ to a given point $-r_2\ei{t_2}$ of the complex plane. If
$\la_3-\la_2=\la_2-\la_1$, then $H$ is the ellipse with centre 0,
half major axis $r_1+r_3$ and half minor axis $\abs{r_1-r_3}$. Note
that an epitrochoid (Ptolemy's epicycle) amounts also to a
hypotrochoid.  Figure~\ref{fig:hypotrochoid} illustrates the
particular case $T(x)=4\mkern1mu\emi{2x}+\ei{t}+\ei{x}$.


We deduce an interval on which $T$ attains its maximum modulus
independently of the moduli of its Fourier coefficients (see
Theorem~\ref{mmpttt}\,$(a)$ for a detailed answer.) We prove in
particular the following result.

\begin{thm}
  \label{unique}
  Let $d=\gcd(\la_2-\la_1,\allowbreak\la_3-\la_2)$ and let
  $\tau$ be the distance of
  \begin{equation}\label{tau}
    \frac{\la_2-\la_3}d\allowbreak t_1
    +\frac{\la_3-\la_1}d\allowbreak t_2
    +\frac{\la_1-\la_2}d\allowbreak t_3
  \end{equation}
  to $2\pi\Z$. The trigonometric trinomial $T$ attains its maximum
  modulus at a unique point modulo $2\pi/d$, with multiplicity 2,
  unless $\tau=\pi$.
\end{thm}

Theorem~\ref{unique} shows that if there are two points of the
hypotrochoid $H$ at maximum distance to $-r_2\ei{t_2}$, it is so only
because $-r_2\ei{t_2}$ lies on an axis of symmetry of $H$.

We obtain a precise description of those trigonometric trinomials that
attain their maximum modulus twice modulo $2\pi/d$: see
Theorem~\ref{mmpttt}\,$(c)$.  Their r\^ole becomes clear by the
following result in convex geometry: they yield the exposed points of
the unit ball of the ambient normed space. Let us first put up the
proper functional analytic framework. Let
$\La=\{\la_1,\la_2,\la_3\}$ be the \emph{spectrum} of the
trigonometric trinomial $T$ and write $\e_\la\colon x\mapsto\ei{\la
  x}$.  Let $\Cont_\La$ be the space of functions spanned by
$\{\e_{\la}:\la\in\La\}$, endowed with the maximum modulus
norm. Recall that a point $P$ of a convex set $K$ is \emph{exposed} if
there is a hyperplane that meets $K$ only in $P$; $P$ is
\emph{extreme} if it is not the midpoint of any two other points of
$K$.
\begin{thm}
  \label{extrpts}
  Let $K$ be the unit ball of the space $\Cont_\La$ and let $P\in
  K$.
  \begin{enumerate}
  \item The point $P$ is an exposed point of $K$ if and only if $P$ is either
    a trigonometric monomial $\ei\alpha\e_\la$ with $\alpha\in\R$ and
    $\la\in\La$ or a trigonometric trinomial that attains its maximum
    modulus, 1, at two points modulo $2\pi/d$. Every linear functional on
    $\Cont_\La$ attains its norm on an exposed point of $K$.
  \item The point $P$ is an extreme point of $K$ if and only if $P$ is either
    a trigonometric monomial $\ei\alpha\e_\la$ with $\alpha\in\R$ and
    $\la\in\La$ or a trigonometric trinomial such that $1-{\abs P}^2$ has
    four zeroes modulo $2\pi/d$, counted with multiplicities.
  \end{enumerate}
\end{thm}

We describe the dependence of the maximum modulus of the trigonometric
trinomial $T$ on the arguments.  The general issue has been studied
for a long time; \cite{li24,sa33} are two early references. In
particular, the following problem has been addressed: see \cite[page~2
and Supplement]{dm90}.
\begin{ep}[Complex Man\-del$'$\-shtam problem]
  To find the minimum of the maximum modulus of a trigonometric
  polynomial with given Fourier coefficient moduli.
\end{ep}
It appeared originally in electrical circuit theory: ``L.~I.
Man\-del$'$\-shtam communicated to me a problem on the phase choice of
electric currents with different frequencies such that the capacity of
the resulting current to blow [the circuit] is minimal'', tells N.~G.
Chebotar\"ev in \cite[p.~396]{ce49}, where he discusses applications
of a formula given in Section~\ref{seccheb} that we would like to
advertise.

Our main theorem solves an elementary case of the complex
Man\-del$'$\-shtam problem.

\begin{thm}
  \label{max-decreasing}
  The maximum modulus of $T$ as defined in~\eqref{tt} 
  is a strictly decreasing function of $\tau$ as defined in Theorem~\ref{unique}. In
  particular, 
  \begin{equation*}
    \min_{t_1,t_2,t_3}\max_x
    \bigabs{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}}
    =\max_{x} 
    \bigabs{\epsilon_1r_1\ei{\la_1x} + \epsilon_2r_2\ei{\la_2x} + \epsilon_3r_3\ei{\la_3x}}
  \end{equation*}
  if $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$ are real signs $+1$ or
  $-1$ such that $\epsilon_i\epsilon_j=-1$, where $i,j,k$ is a
  permutation of $1,2,3$ such that the power of 2 in $\la_i-\la_j$ is
  greater than the power of 2 in $\la_i-\la_k$ and in
  $\la_k-\la_j$.
\end{thm}
This shows that the maximum modulus is minimal when the phases are
chosen in opposition, independently of the intensities $r_1$, $r_2$
and $r_3$.

The decrease of the maximum modulus of~\eqref{tt} may be bounded as
shown in the next result.

\begin{thm}
  \label{how-dec}
  Let $d$ and $\tau$ be defined as in Theorem~\ref{unique}. Suppose
  that $\la_2$ is between $\la_1$ and $\la_3$.  The quotient of the
  maximum modulus of $T$ by $\bigabs{r_1+r_2\ei{\tau
      d/\abs{\la_3-\la_1}}+r_3}$ is a strictly increasing function of
  $\tau$ unless $r_1:r_3=\abs{\la_3-\la_2}:\abs{\la_2-\la_1}$, in
  which case it is constantly equal to 1.
\end{thm}

When $r_1:r_3=\abs{\la_3-\la_2}:\abs{\la_2-\la_1}$, the hypotrochoid
$H$ with equation~\eqref{hypo} is a \emph{hypocycloid} with
$\abs{\la_3-\la_1}/d$ cusps: the rolling point is \emph{on} the
rolling circle. Note that an epicycloid amounts also to an hypocycloid. 
%If ..., then $H$ is also anepicycloid. 
Figure~\ref{fig:2} illustrates the particular case
$T(x)=(1/3)\emi{2x}+\ei{t}+(2/3)\ei{x}$.
\begin{figure}[htbm]
  \centering
  \begin{pspicture}(-2.95,-2.55)(2.95,2.507)
    \parametricplot[plotstyle=ccurve,linewidth=.4pt]{0}{360}{t cos 5
      mul 3 div t 2 mul cos 5 mul 6 div add t sin 5 mul 3 div t 2 mul
      sin 5 mul 6 div sub}
    \pscircle[linewidth=.4pt](0,0){2.507}
    \psline[linewidth=.7pt,linestyle=dotted,dotsep=1.5pt](-2.5,0)(2.5,0)
    \psline[linewidth=.4pt,linestyle=dashed,dash=2pt
    2pt](2.5,0)(-1.25,-2.16506)(-1.25,2.16506)
    \rput[l](-2.95,0){\small$-1$}
    \rput[Bl](-2.1,-2.55){\small$-\ei{\pi/3}$}\vspace{2cm}
  \end{pspicture}
  \caption{The unit circle, the deltoid $H$ with equation
    $z=(1/3)\emi{2x}+(2/3)\ei{x}$, the segment from $-1$ to the unique
    point on $H$ at maximum distance and the segments from
    $-\ei{\pi/3}$ to the two points on $H$ at maximum distance.}
  \label{fig:2}
\end{figure}

We may deduce from Theorem~\ref{how-dec} a less precise but handier
inequality.

\begin{thm}
  \label{th-4}
  Let $d$ and $\tau$ be defined as in Theorem~\ref{unique}. Let
  \begin{equation}
    \label{D}
    D=\frac
    {\max(\abs{\la_2-\la_1},\abs{\la_3-\la_2},\abs{\la_3-\la_1})}
    {\gcd(\la_2-\la_1,\la_3-\la_1)}
  \end{equation}
  be the quotient of the diameter of $\La$ by $d$.
Let $t'_1$, $t'_2$ and $t'_3$ be another three real numbers and define
  correspondingly $\tau'$.  If $\tau>\tau'$, then
  \begin{multline*}
    \max_x\bigabs{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}}\\
    \ge\frac{\cos(\tau /2D)}{\cos(\tau'/2D)}
    \max_x\bigabs{r_1\ei{(t'_1+\la_1x)}+r_2\ei{(t'_2+\la_2x)}+r_3\ei{(t'_3+\la_3x)}}
  \end{multline*}
  with equality if and only if
  $r_1:r_2:r_3=\abs{\la_3-\la_2}:\abs{\la_3-\la_1}:\abs{\la_2-\la_1}$.
\end{thm}
Figure~\ref{fig:3} illustrates the inequalities obtained in
Theorems~\ref{how-dec} and~\ref{th-4} for the particular case
$T(x)=4\emi{2x}+\ei{t}+\ei{x}$, as in Figure~\ref{fig:hypotrochoid}.
\begin{figure}[htbm]
  \psset{xunit=3cm,yunit=3cm,linewidth=.4pt}
  \begin{pspicture}(-1.6,4.85)(1.6,6.1) 
    \savedata{\V}[-.20944 5.9852 -.15708 5.9917 -.10472 5.9963
      -.052361 5.9991 0 6 .052361 5.9991 .10472 5.9963 .15708 5.9917
      .20944 5.9852 .2618 5.9768 .31416 5.9666 .36653 5.9546 .41887
      5.9407 .47124 5.9251 .52361 5.9077 .57595 5.8885 .62832 5.8676
      .68069 5.8449 .73303 5.8206 .7854 5.7946 .83777 5.7669 .89011
      5.7376 .94248 5.7068 .99485 5.6743 1.0472 5.6404]
    \dataplot[plotstyle=curve]{\V}
    \savedata{\U}[1.0472 5.6404 1.0996 5.6743 1.1519 5.7068 1.2043
      5.7376 1.2566 5.7669 1.309 5.7946 1.3613 5.8206 1.4137 5.8449
      1.4661 5.8676 1.5184 5.8885 1.5708 5.9077]
    \dataplot[plotstyle=curve]{\U}
    \psline[linestyle=solid]{->}(0,4.95)(0,6.1)
    \psline[linestyle=solid]{->}(-0.20944,5)(1.6,5)
    \rput(0,4.88){\small0} \rput[b](0,6.12){\small$m$}
    \rput[r](-.23,5){\small5} \rput[r](1.66,5){\small$t$}
    \rput(1.0472,4.88){\small$\pi/3$}
    \rput[l](-1.6,5.97){\small$m=\max\limits_x|4\emi{2x}+\ei{t}+\ei{x}|$}
    \rput[l](1.07,5.57){\small$m=|4+\ei{t}+1|$}
    \rput[l](1.07,5.2){\small$m=(4+1+1)\cos(t/2)$}
    \psline[linestyle=dashed](1.0472,4.95)(1.0472,6.1)
    \psplot[plotstyle=curve,linestyle=dashed,dash=2pt
    2pt]{0}{1.0472}{x 57.29578 mul cos 10 mul 26 add sqrt}
    \psplot[plotstyle=curve,linestyle=dotted,linewidth=.7pt,dotsep=1.5pt]{0}{1.0472}{x
      57.29578 mul 2 div cos 6 mul}
  \end{pspicture}
  \caption{Let $H$ be the hypotrochoid with equation
    $z=4\emi{2x}+\ei{x}$. This plot shows the maximum distance $m$ of
    points $z\in H$ to the point $-\ei{t}$ and the two estimates of
    this maximum distance provided by
    Theorems~\ref{how-dec} and~\ref{th-4} for $t\in[0,\pi/3]$.}
  \label{fig:3}
\end{figure}

If we choose $\tau'=0$ in Theorem~\ref{th-4}, we get the solution to an
elementary case of the following extremal problem.
\begin{ep}
  To find the minimum of the maximum modulus of a trigonometric
  polynomial with given spectrum, Fourier coefficient arguments and
  moduli sum.
\end{ep}


\begin{thm}
  \label{th-3}
  Let $\tau$ be defined as in Theorem~\ref{unique} and $D$ be given by \eqref{D}. Then
  \begin{equation*}
    \frac
    {\max_x\bigabs{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}}}
    {r_1+r_2+r_3}
    \ge\cos(\tau /2D)
  \end{equation*}
  with equality if and only if $\tau=0$ or
  $r_1:r_2:r_3=\abs{\la_3-\la_2}:\abs{\la_3-\la_1}:\abs{\la_2-\la_1}$.
\end{thm}

The dependence of the maximum modulus of~\eqref{tt} on the arguments
may also be expressed as properties of relative multipliers.  Given
three real numbers $t_1$, $t_2$ and $t_3$, the linear operator on
$\Cont_\La$ defined by $\e_{\la_j}\mapsto\ei{t_j}\e_{\la_j}$ is a
\emph{unimodular relative Fourier multiplier}: it multiplies each
Fourier coefficient of elements of $\Cont_\La$ by a fixed
unimodular number; let us denote it by $(t_1,t_2,t_3)$. Consult
\cite{ha87} for general background on relative multipliers.

\begin{thm}
  \label{multiplier}
  The unimodular relative
  Fourier multiplier $(t_1,t_2,t_3)$ has norm
  \begin{equation*}
    \cos\bigl((\pi-\tau)/2D\bigr)\big/\cos(\pi/\allowbreak2D),
  \end{equation*}
  where $\tau$ is defined as in Theorem~\ref{unique} and $D$ is given by~\eqref{D}, and attains its norm exactly
  at functions of the form
  \begin{equation*}
    r_1\ei{(u_1+\la_1x)}+r_2\ei{(u_2+\la_2x)}+r_3\ei{(u_3+\la_3x)}
  \end{equation*}
  with $r_1:r_2:r_3=\abs{\la_3-\la_2}:\abs{\la_3-\la_1}:\abs{\la_2-\la_1}$ and
  \begin{equation*}
    \frac{\la_2-\la_3}du_1+\frac{\la_3-\la_1}du_2+\frac{\la_1-\la_2}du_3=\pi\mod{2\pi}.
  \end{equation*}
\end{thm}

The maximum of the norm of unimodular relative Fourier multipliers is
the \emph{complex unconditional constant} of the canonical basis
$(\e_{\la_1},\e_{\la_2},\e_{\la_3})$ of $\Cont_\La$. As
\begin{equation*}
  r_1+r_2+r_3=\max_x\bigabs{r_1\ei{\la_1x}+r_2\ei{\la_2x}+r_3\ei{\la_3x}},
\end{equation*}
this constant is the minimal constant $C$ such that
\begin{equation*}
  r_1+r_2+r_3\le C\max_x\bigabs{r_1\ei{(u_1+\la_1x)}+r_2\ei{(u_2+\la_2x)}+r_3\ei{(u_3+\la_3x)}};
\end{equation*}
it is therefore the \emph{Sidon constant} of $\La$. It is also the
solution to the following extremal problem.
\begin{ep}[Sidon constant problem]
  To find the minimum of the maximum modulus of a trigonometric polynomial
  with given spectrum and Fourier coefficient moduli sum.
\end{ep}

Setting $\tau=\pi$ in Theorem~\ref{multiplier}, we obtain the following result.

\begin{cor}
  The Sidon constant of $\La$ is $\sec(\pi/\allowbreak2D)$, where
  $D$ is given by~\eqref{D}. It is attained exactly at functions of
  the form given in Theorem~\ref{multiplier}.
\end{cor}

Finally, we would like to stress that each of the above results gives
rise to open questions if the set $\La$ is replaced by \emph{any}
set of four integers.

Let us now give a brief description of this article. In
Sections~\ref{sec2} and~\ref{sec2bis}, we use carefully the invariance
of the maximum modulus under rotation, translation and conjugation to
reduce the arguments $t_1$, $t_2$ and $t_3$ of the Fourier
coefficients of the trigonometric trinomial $T$ to the variable
$\tau$.  Section~\ref{sec3} shows how to further reduce this study to
the trigonometric trinomial
\begin{equation}
  \label{Tred}
  r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}
\end{equation}
with $k$ and $l$ positive coprime integers and $t\in[0,\pi/(k+l)]$. In
Section~\ref{secloc}, we prove that~\eqref{Tred} attains its maximum
modulus for $x\in[-t/k,t/l]$. Section~\ref{sec4} studies the
variations of the modulus of~\eqref{Tred} for $x\in[-t/k,t/l]$: it
turns out that it attains its absolute maximum only once on that
interval. This yields Theorem~\ref{unique}.  Section~\ref{secmm}
restates the results of the two previous sections for a general
trigonometric trinomial $T$. Section~\ref{sec:exp} is dedicated to the
proof of Theorem~\ref{extrpts}. In Section~\ref{seccheb}, we compute
the directional derivative of the maximum modulus of~\eqref{Tred} with
respect to the argument $t$ and prove Theorems~\ref{max-decreasing},
\ref{how-dec}, \ref{th-4} and~\ref{th-3}. In Section~\ref{secnm}, we
prove Theorem~\ref{multiplier} and show how to lift unimodular
relative Fourier multipliers to operators of convolution with a linear
combination of two Dirac measures.  Section~\ref{secsidon} replaces
our computation of the Sidon constant in a general context; it
describes the initial motivation for this research.

Part of these results appeared previously, with a different proof, in
\cite[Cha\-pter II.10]{ne99} and in \cite{ne01b}.

\begin{Not}
  Throughout this article, $\la_1$, $\la_2$ and $\la_3$ are three
  pairwise distinct integers, $\La$ is the set $\{\la_1,\la_2,\la_3\}$ and
  $d=\gcd(\la_2-\la_1,\allowbreak\la_3-\la_2)$. If $\la$ is an
  integer, $\e_\la$ is the function $x\mapsto\ei{\la x}$ of the real
  variable $x$.  A \emph{trigonometric polynomial} is a linear
  combination of functions $\e_\la$; it is a \emph{monomial},
  \emph{binomial} or \emph{trinomial} if this linear combination has
  one, two or three nonzero coefficients, respectively. The normed
  space $\Cont_\La$ is the three-dimensional space of complex
  functions spanned by $\e_\la$ with $\la\in\La$, endowed with the
  maximum modulus norm. The Dirac measure $\delta_x$ is the linear
  functional $T\mapsto T(x)$ of evaluation at $x$ on the space of
  continuous functions. Given three real numbers $t_1$, $t_2$ and
  $t_3$, the linear operator on $\Cont_\La$ defined by
  $\e_{\la_j}\mapsto\ei{t_j}\e_{\la_j}$ is a \emph{unimodular relative
    Fourier multiplier} denoted by $(t_1,t_2,t_3)$.
\end{Not}

\section{Isometric relative Fourier multipliers}
\label{sec2}

\noindent The r\^ole of Quantity~\eqref{tau} is explained by the following lemma.

\begin{lem}
  \label{isom}
  Let $t_1$, $t_2$ and $t_3$ be real numbers. The unimodular relative
  Fourier multiplier $M=(t_1,\allowbreak t_2,\allowbreak t_3)$ is an
  isometry on $\Cont_\La$ if and only if
  \begin{equation}
    \label{th}
    \frac{\la_2-\la_3}dt_1
    +\frac{\la_3-\la_1}dt_2
    +\frac{\la_1-\la_2}dt_3
    \in2\pi\Z.
  \end{equation}
  Then it is a unimodular multiple of a translation: there are real
  numbers $\alpha$ and $v$ such that $Mf(x)=\ei\alpha f(x-v)$ for all
  $f\in\Cont_\La$ and all $x\in\R$.
\end{lem}

\begin{proof}
  If $M$ is a unimodular multiple of a
  translation by a real number $v$, then
  \begin{equation*}
    \bigabs{r_1\ei{(t_1+\la_1v)}+r_2\ei{(t_2+\la_2v)}+r_3\ei{(t_3+\la_3v)}}=r_1+r_2+r_3,
  \end{equation*}
  which holds if and only if
  \begin{equation}
    \label{thx}
    t_1+\la_1v=t_2+\la_2v=t_3+\la_3v\quad\text{modulo $2\pi$.}
  \end{equation}
  There is a $v$ satisfying~\eqref{thx} if and only if
  Equation~\eqref{th} holds as~\eqref{thx} means that there exist
  integers $a_1$ and $a_3$ such that
  \begin{equation*}
    v
    =\frac{t_2-t_1+2\pi a_1}{\la_1-\la_2}
    =\frac{t_2-t_3+2\pi a_3}{\la_3-\la_2}.
  \end{equation*}
  If $t_1$, $t_2$ and $t_3$ are three real numbers satisfying~\eqref{th}, let
  $v$ be such that~\eqref{thx} holds. Then
  \begin{multline*}
    {r_1\ei{(t_1+u_1+\la_1 x)}
      +r_2\ei{(t_2+u_2+\la_2 x)}
      +r_3\ei{(t_3+u_3+\la_3 x)}}\\
    =\ei{(t_2+\la_2v)}\bigl({r_1\ei{(u_1+\la_1(x-v))}+r_2\ei{(u_2+\la_2(x-v))}+r_3\ei{(u_3+\la_3(x-v))}}\bigr)
  \end{multline*}
  for all real numbers $u_1$, $u_2$, $u_3$ and $x$.
\end{proof}



\section[The arguments of the Fourier coefficients of a trinomial]{%
The arguments of the Fourier coefficients of a trigonometric trinomial}

\label{sec2bis}




\noindent We have used a translation and a rotation to reduce the three
arguments of the Fourier coefficients of a trigonometric trinomial to
just one variable. Use of the involution $\overline{f(-x)}$ of
$\Cont_\La$ allows us to restrict even further the domain of that
variable.

\begin{lem}
  \label{upto}
  Let $t_1$, $t_2$ and $t_3$ be real numbers and let $\tilde t_2$ be the representative of
  \begin{equation}
    \label{tilde}
    \frac{\la_2-\la_3}{\la_3-\la_1}\allowbreak t_1
    +t_2
    +\frac{\la_1-\la_2}{\la_3-\la_1}\allowbreak t_3 
  \end{equation}
  modulo $2\pi/\abs{\la_3-\la_1}$ in $\bigiod[-\pi d/\abs{\la_3-\la_1},\pi d/\abs{\la_3-\la_1}]$.
  \begin{enumerate}
  \item There are real numbers $\alpha$ and $v$ such that
    \begin{equation}
      \label{nett}
      {r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}}
      =\ei\alpha\bigl({r_1\ei{\la_1(x-v)}+r_2\ei{(\tilde
          t_2+\la_2(x-v))}+r_3\ei{\la_3(x-v))}}\bigr)
    \end{equation}
    for all $x$.
  \item Let $t=\abs{\tilde t_2}$ be the distance of~\eqref{tilde} to $(2\pi
    d/\abs{\la_3-\la_1})\Z$. There is a sign $\eps\in\{+1,-1\}$ such that
    \begin{equation*}
      \bigabs{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}}
      =\bigabs{r_1\ei{\la_1\eps(x-v)}+r_2\ei{(t+\la_2\eps(x-v))}+r_3\ei{\la_3\eps(x-v))}}
    \end{equation*}
    for all $x$.
  \end{enumerate}
\end{lem}

\begin{proof}
  $(a)$. The argument $\tilde t_2$ is chosen so that the relative multiplier
  $(t_1,\allowbreak t_2-\tilde t_2,\allowbreak t_3)$ is an isometry. 

  $(b)$. If $\tilde t_2$ is negative, take the conjugate under the modulus of
  the right hand side in~\eqref{nett}.
\end{proof}

\begin{rem}\label{perio}
  This proves the following periodicity formula:
  \begin{equation*}
    \bigabs{r_1\ei{\la_1x} +r_2\ei{(t+2\pi d/(\la_3-\la_1)+\la_2x)}
      +r_3\ei{\la_3x}}
    =\bigabs{r_1\ei{\la_1(x-v)} +r_2\ei{(t+\la_2(x-v))}
      +r_3\ei{\la_3(x-v)}}
  \end{equation*}
  for all $x$ and $t$, where $v$ satisfies $\la_1v=2\pi
  d/(\la_3-\la_1)+\la_2v=\la_3v$ modulo $2\pi$, that is
  \begin{equation*}
    v=\frac{2m\pi}{\la_3-\la_1}\quad\text{with $m$ an inverse of $\frac{\la_3-\la_2}d$
      modulo $\frac{\la_3-\la_1}d$.}
  \end{equation*}
\end{rem}

\section{The frequencies of a trigonometric trinomial}\label{sec3}

\noindent We may suppose without loss of generality that $\la_1<\la_2<\la_3$.
Let $k=(\la_2-\la_1)/\allowbreak d$ and $l=(\la_3-\la_2)/d$. Then
\begin{equation*}
r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}
=\ei{\la_2x}(r_1\ei{(t_1-k(dx))}+r_2\ei{t_2}+r_3\ei{(t_3+l(dx))})
\end{equation*}
This defines an isometry between
$\Cont_\La$ and $\Cont_{\{-k,0,l\}}$ and shows that $\Cont_\La$ is normed by the maximum modulus norm
on $[0,2\pi/d\mathclose[$. With Lemma~\ref{upto}\,$(b)$, this shows that 
% \begin{equation*}
%   \bigabs{r_1\ei{\la_1x}
%     +r_2\ei{(t+\la_2x)}
%     +r_3\ei{\la_3x}}
%   =\bigabs{r_1\emi{k(dx)}
%     +r_2\ei{t}
%     +r_3\ei{l(dx)}}.
% \end{equation*}
a homothety by $d^{-1}$  allows us to restrict our study to the
function
\begin{equation*}
  f(t,x)
  ={\bigabs{r_1\emi{kx}
      +r_2\ei{t}
      +r_3\ei{lx}}}^2
\end{equation*}
for $x\in\R$ with $k$ and $l$ two positive coprime numbers and
$t\in\bigl[0,\pi/(k+l)\bigr]$.  We have
\begin{equation}
  {f(-t,x)}
  ={f(t,-x)}
  \label{pair}
\end{equation}
and Remark~\ref{perio} shows that
\begin{equation}
  {f\bigl(t+2\pi/(k+l),x\bigr)}
  ={f(t,x-2m\pi/(k+l))}
  \label{per}
\end{equation}
for all $x$ and $t$, where 
$m$ is the inverse of $l$ modulo $k+l$.
In particular, if $t=\pi/(k+l)$, we have the symmetry relation
\begin{equation}
  \label{symmetry}
  {f\bigl(\pi/(k+l),x\bigr)}
  ={f\bigl(\pi/(k+l),2m\pi/(k+l)-x\bigr)}\text.
\end{equation}



\section{Location of the maximum point}
\label{secloc}



\noindent The purpose of our first proposition is to deduce the existence of a small interval on
which a trigonometric trinomial attains its maximum modulus. Note that
a trigonometric binomial attains its maximum modulus at a point that
depends only on the phase of its coefficients:
\begin{itemize}
\item $\bigabs{r_1\emi{kx}+r_2\ei{t}}$ attains its maximum at $-t/k$
  independently of $r_1$ and $r_2$,
\item $\bigabs{r_1\emi{kx}+r_3\ei{lx}}$ attains its maximum at 0
  independently of $r_1$ and $r_3$,
\item $\bigabs{r_2\ei{t}+r_3\ei{lx}}$ attains its maximum at $t/l$
  independently of $r_2$ and $r_3$.
\end{itemize}
The next proposition shows that if
the point at which a trigonometric trinomial attains its maximum
modulus changes with the modulus of its coefficients, it changes
very little; we get bounds for this point that are independent of
the intensities.


\begin{prp}
  \label{loc}
  Let $k$ and $l$ be positive coprime integers. Let $r_1$, $r_2$ and
  $r_3$ be three positive real numbers. Let
  $t\in\bigl[0,\pi/(k+l)\bigr]$.  Let
  \begin{equation*}
    f(x)={\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}}^2
  \end{equation*}
  for $x\in\R$.
  \begin{enumerate}
  \item The function $f$ attains its absolute maximum in the
    interval $[-t/k,\allowbreak t/l]$.
  \item If $ f$ attains its absolute maximum at a point $y$ outside of
    $[-t/k,t/l]$ modulo $2\pi$, then $t=\pi/(k+l)$ and
    $2m\pi/(k+l)-y$ lies in $[-t/k,t/l]$ modulo $2\pi$, where $m$ is
    the inverse of $l$ modulo $k+l$.
  \end{enumerate}
\end{prp}

\begin{proof}
  $(a)$. We have
  \begin{equation}
    \label{f(x)carre}
    {f(x)}=r_1^2+r_2^2+r_3^2+2\cdot
    \bigl(r_1r_2\cos(t+kx)+r_1r_3\cos\bigl((k+l)x\bigr)+r_2r_3\cos(t-lx)\bigr).
  \end{equation}
  Let us prove that ${f}$ attains its absolute maximum on
  $[-t/k,\allowbreak t/l]$.  Let $y$ be outside of $[-t/k,t/l]$ modulo
  $2\pi$. Let $I$ be the set of all $x\in[-t/k,t/l]$ such that
  \begin{equation*}
    \left\{
      \begin{aligned}
        \cos(t+kx)&\ge\cos(t+ky)\\
        \cos\bigl((k+l)x\bigr)&\ge\cos\bigl((k+l)y\bigr)\\
        \cos(t-lx)&\ge\cos(t-ly).
      \end{aligned}
    \right.
  \end{equation*}
  Note that if $x\in[-t/k,t/l]$, then
  \begin{equation*}
    \left\{
      \begin{aligned}
        t+kx&\in[0,(k+l)t/l]\\ 
        (k+l)x&\in[-(k+l)t/k,(k+l)t/l]\\ 
        t-lx&\in[0,(k+l)t/k],
      \end{aligned}
    \right.
  \end{equation*}
  and that $(k+l)t/k$, $(k+l)t/l\in[0,\pi]$. Let
  \begin{itemize}
  \item $\alpha$ be the distance of $t/k+y$ to $(2\pi/k)\Z$,
  \item $\beta$ be the distance of $y$ to $\bigl(2\pi/(k+l)\bigr)\Z$,
  \item $\gamma$ be the distance of $t/l-y$ to $(2\pi/l)\Z$.
  \end{itemize}
  Then
  \begin{equation}
    \label{II}
    I=[-t/k,t/l]\cap
    [-t/k-\alpha,-t/k+\alpha]\cap
    [-\beta,\beta]\cap
    [t/l-\gamma,t/l+\gamma].
  \end{equation}
  Let us check that $I$ is the nonempty interval
  \begin{equation}\label{I}
    I=\bigl[\max(-t/k,-\beta,t/l-\gamma),\min(t/l,-t/k+\alpha,\beta)\bigr].
  \end{equation}
  In fact, we have the following triangular inequalities:
  \begin{itemize}
  \item $-\beta\le-t/k+\alpha$ because $t/k$ is the
    distance of $(t/k+y)-y$ to $(2\pi/k(k+l))\Z$;
  \item $t/l-\gamma\le-t/k+\alpha$ because $t/l+t/k$ is the distance of
    $(t/k+y)+(t/l-y)$ to $(2\pi/kl)\Z$;
  \item $t/l-\gamma\le\beta$ because
    $t/l$ is the distance of
    $(t/l-y)+y$ to $(2\pi/l(k+l))\Z$.
  \end{itemize}
  The other six inequalities that are necessary to deduce~\eqref{I}
  from~\eqref{II} are obvious. 

  $(b)$. We have proved in $(a)$ that there is an $x\in[-t/k,t/l]$ such
  that $\cos(t+kx)\ge\cos(t+ky)$,
  $\cos\bigl((k+l)x\bigr)\ge\cos\bigl((k+l)y\bigr)$ and
  $\cos(t-lx)\ge\cos(t-ly)$. In fact, at least one of these inequalities is
  strict unless there are signs $\delta,\eps,\eta\in\{-1,1\}$ such that
  $t+kx=\delta(t+ky)$, $t-lx=\eps(t-ly)$ and $(k+l)x=\eta(k+l)y$
  modulo $2\pi$. Two out of these three signs are equal and the corresponding
  two equations imply the third one with the same sign. This system is
  therefore equivalent to
  \begin{equation*}
    \left\{
      \begin{aligned}
        k(x-y)&=0\\
        l(x-y)&=0
      \end{aligned}
    \right.
    \quad\textrm{or}\quad
    \left\{
      \begin{aligned}
        k(x+y)&=-2t\\
        l(x+y)&=2t
      \end{aligned}
    \right.
  \end{equation*}
  modulo $2\pi$. The first pair of equations yields $x=y$ modulo
  $2\pi$ because $k$ and $l$ are coprime. Let $m$ be an inverse of $l$
  modulo $k+l$; then the second pair of equations is equivalent to
  \begin{equation*}
    \left\{
      \begin{aligned}
        2(k+l)t&=0\\
        x+y&=2mt
      \end{aligned}
    \right.
  \end{equation*}
  modulo $2\pi$. Therefore $g$ does not attain its absolute maximum at
  $y$ unless $t=\pi/(k+l)$ and $2m\pi/(k+l)-y\in[-t/k,t/l]$.
\end{proof}

\begin{rem}
  This proposition is a complex counterpart to Lemma~$2.1.\,i)$ in
  \cite{re95}, where cosine trinomials are investigated.
\end{rem}

\section{Uniqueness of the maximum point}
\label{sec4}

\noindent Note that
\begin{align*}
  r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}
  &=r_3\emi{l(-x)}+r_2\ei{t}+r_1\ei{k(-x)}\\
  &=r'_1\emi{k'x'}+r_2\ei{t}+r'_3\ei{l'x'}
\end{align*}
with $r'_1=r_3$, $r'_3=r_1$, $k'=l$, $l'=k$ and $x'=-x$.
We may therefore suppose without loss of generality that $kr_1\le lr_3$.

Our second proposition studies the points at which a trigonometric trinomial
attains its maximum modulus. Note that if $k=l=1$, the derivative of
${\abs{f}}^2$ has at most 4 zeroes, so that the modulus of $f$ has at most
two maxima and attains its absolute
maximum in at most two points. Proposition~\ref{prp-2} shows that this is true
in general, and that if it may attain its absolute maximum in two points, it
is so only because of the symmetry given by~\eqref{symmetry}.


\begin{prp}
  \label{prp-2}
  Let $k$ and $l$ be positive coprime integers. Let $r_1$, $r_2$ and
  $r_3$ be three positive real numbers such that $kr_1\le lr_3$. Let
  $t\in\iog[0,\pi/(k+l)]$. Let
  \begin{equation*}
    f(x)={\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}}^2
  \end{equation*}
  for $x\in[-t/k,t/l]$.
  \begin{enumerate}
  \item There is a point $x^*\in[0,t/l]$ such that $\mathrm df/\mathrm dx>0$
    on $\io[-t/k,x^*]$ and $\mathrm df/\mathrm dx<0$ on $\io[x^*,t/l]$.
  \item There are three cases:
    \begin{enumerate}
    \item ${f}$ attains its absolute maximum at 0 if and only if
      $kr_1=lr_3$;
    \item ${f}$ attains its absolute maximum at $t/l$ if and only if
      $l=1$, $t=\pi/(k+1)$ and $k^2r_1r_2+(k+1)^2r_1r_3-r_2r_3\le0$;
    \item otherwise, ${f}$ attains its absolute maximum in
      $\io[0,t/l]$.
    \end{enumerate}
  \item The function $f$ attains its absolute maximum with
    multiplicity 2 unless $l=1$, $t=\pi/(k+1)$ and
    $k^2r_1r_2+(k+1)^2r_1r_3-r_2r_3=0$, in which case it attains its
    absolute maximum at $\pi/(k+1)$ with multiplicity 4.
  \end{enumerate}
\end{prp}
\begin{proof}
  $(a)$. By Proposition~\ref{loc}, the derivative of $f$ has a zero in
  $[-t/k,t/l]$. Let us study the sign of this derivative.
  Equation~\eqref{f(x)carre} yields
  \begin{equation}\label{df(x)}
    \frac12\dfrac{\mathrm d{f}}{\mathrm dx}(x)
    =-kr_1r_2\sin(t+kx)-(k+l)r_1r_3\sin\bigl((k+l)x\bigr)+lr_2r_3\sin(t-lx).
  \end{equation}

  We wish to compare $\sin(t+kx)$ with $\sin(t-lx)$: note that
  \begin{equation*}
    \sin(t+kx)-\sin(t-lx)
    =2\sin\bigl((k+l)x/2\bigr)
    \cos\bigl(t+(k-l)x/2\bigr),
  \end{equation*}
  and that if $x\in[-t/k,t/l]$, then
  \begin{gather*}
    -\pi/2\le-\pi/2k\le-(k+l)t/2k\le(k+l)x/2\le(k+l)t/2l\le\pi/2l\le\pi/2\\
    0\le t+(k-l)x/2\le~
    \begin{cases}
      t+(l-k)t/2k=(k+l)t/2k&\text{if $k\le l$}\\
      t+(k-l)t/2l=(k+l)t/2l&\text{if $l\le k$}
    \end{cases}
    ~\le\pi/2.
  \end{gather*}

  Suppose that $x\in\iod[-t/k,0]$: then it follows that
  $\sin(t+kx)\le\sin(t-lx)$ and also $\sin\bigl((k+l)x\bigr)\le0$, with
  equality if and only if $k=1$ and $-x=t=\pi/(1+l)$. This yields
  with $kr_1\le lr_3$ that
  \begin{equation}
    \label{1}
    \frac12\dfrac{\mathrm d{f}}{\mathrm dx}(x)
    \ge-(k+l)r_1r_3\sin\bigl((k+l)x\bigr)\ge0
  \end{equation}
  with equality if and only if $k=1$ and $-x=t=\pi/(1+l)$. 

  Suppose that $x\in[0,t/l]$. If $l\ge2$, then
  \begin{equation*}
    \left\{
      \begin{aligned}
        t+kx&\in[t,(k+l)t/l]\subset[t,\pi/2]\\ 
        (k+l)x&\in[0,(k+l)t/l]\subset[t,\pi/2]\\ 
        t-lx&\in[0,t]\subset[0,\pi/3],
      \end{aligned}
    \right.
  \end{equation*}
  so that the second derivative of ${f}$ is strictly negative on
  $[0,t/l]$: its derivative is strictly decreasing on this interval
  and $(a)$ is proved. If $l=1$, let $g(x)=f(t-x)$ for $x\in[0,t]$: we
  have to prove that there is a point $x^*$ such that $\mathrm
  dg/\mathrm dx>0$ on $\io[0,x^*]$ and $\mathrm dg/\mathrm dx<0$ on
  $\io[x^*,t]$. We already know that
  \begin{math}
    (\mathrm dg/\mathrm dx)(0)\ge0
  \end{math}
  and that
  \begin{math}
    \mathrm dg/\mathrm dx
  \end{math}
  has a zero on $[0,t]$.  Put $\alpha=(k+1)t$: then
  \begin{equation*}
    \frac12\dfrac{\mathrm d{g}}{\mathrm dx}(x)
    =kr_1r_2\sin(\alpha-kx)
    +(k+1)r_1r_3\sin\bigl(\alpha-(k+1)x\bigr)
    -r_2r_3\sin x
  \end{equation*}
  and it suffices to prove that
  \begin{equation}
    \label{g}
    \dfrac{1}{2\sin x}\dfrac{\mathrm d{g}}{\mathrm dx}(x)
    =kr_1r_2\frac{\sin(\alpha-kx)}{\sin x}
    +(k+1)r_1r_3\frac{\sin\bigl(\alpha-(k+1)x\bigr)}{\sin x}
    -r_2r_3
  \end{equation}
  is a strictly decreasing function of $x$ on
  $\iog[0,{\alpha/(k+1)}]$.  Let us study the sign of
  \begin{equation*}
    \frac{\mathrm d}{\mathrm dx}\frac{\sin(\alpha-kx)}{\allowbreak\sin x}
    =\frac{-k\cos(\alpha-kx)\sin x-\sin(\alpha-kx)\cos x}{\sin^2x}
  \end{equation*}
  for $\alpha\in\iog[0,\pi]$ and $x\in\iog[0,{\alpha/k}]$. If $k=1$,
  then
  \begin{equation*}
    -k\cos(\alpha-kx)\sin x-\sin(\alpha-kx)\cos x=-\sin\alpha\le0
  \end{equation*}
  and the inequality is strict unless $\alpha=\pi$. Let us prove by
  induction on $k$ that
  \begin{equation*}
    k\cos(\alpha-kx)\sin x+\sin(\alpha-kx)\cos x>0
\end{equation*}
  for all $k\ge2$, $\alpha\in\iog[0,\pi]$ and
  $x\in\iog[0,{\alpha/k}]$. This will complete the proof of $(a)$. Let
  $k\ge1$ and $x\in\iog[0,{\alpha/(k+1)}]$. Then
  \begin{multline*}
    (k+1)\cos\bigl(\alpha-(k+1)x\bigr)\sin x
    +\sin\bigl(\alpha-(k+1)x\bigr)\cos x\\
    \begin{aligned}
     &=\begin{aligned}[t]
        (k+1)\cos(\alpha-kx)\cos x\sin x+(k+1)\sin(\alpha-kx)\sin^2 x\\
        +\sin(\alpha-kx)\cos^2 x-\cos(\alpha-kx)\sin x\cos x&
      \end{aligned}\\
      &=\begin{aligned}[t]
        \bigl(k\cos(\alpha-kx)\sin x+\sin(\alpha-kx)\cos x\bigr)\cos x\\
        +(k+1)\sin(\alpha-kx)\sin^2 x&
      \end{aligned}\\
      &\ge(k+1)\sin(\alpha-kx)\sin^2 x>0
    \end{aligned}
  \end{multline*}

  $(b)$. 1. By Proposition~\ref{loc} and $(a)$, $f$ attains its
  absolute maximum at 0 if and only if 0 is a critical point for
  $f$. We have
  \begin{equation*}
    \frac12\dfrac{\mathrm d{f}}{\mathrm dx}(0)
    =(lr_3-kr_1)r_2\sin t\ge0
  \end{equation*}
  and equality holds if and only if $kr_1=lr_3$. 

  2. We have
  \begin{equation*}
    \frac12\dfrac{\mathrm d{f}}{\mathrm dx}(t/l)
    =\bigl(-kr_1r_2-(k+l)r_1r_3\bigr)\sin\bigl((k+l)t/l\bigr)\le0
  \end{equation*}
  and equality holds if and only if $l=1$ and $t=\pi/(k+1)$.  Let
  $l=1$ and $t=\pi/(k+1)$ and let us use the notation introduced in
  the last part of the proof of $(a)$: we need to characterise the case that
  $g$ has a maximum at 0. As $\alpha=\pi$ and
  \begin{equation}\label{d2g}
    \frac1{2\sin x}\dfrac{\mathrm d{g}}{\mathrm dx}(x)
    =k^2r_1r_2+(k+1)^2r_1r_3-r_2r_3+o(x)
  \end{equation}
  is a strictly decreasing function of $x$ on $\iog[0,{\pi/(k+1)}]$,
  $g$ has a maximum at 0 if and only if
  $k^2r_1r_2+(k+1)^2r_1r_3-r_2r_3\le0$.

  $(c)$. If $l\ge2$, then the second derivative of $f$ is strictly
  negative on $[0,t/l]$. If $l=1$, then the derivative of~\eqref{g} is
  strictly negative on $\iog[0,\alpha/(k+1)]$: this yields that the
  second derivative of $g$ can only vanish at 0. By $(b)\,2.$, $g$ has
  a maximum at 0 only if $t=\pi/(k+1)$; then
  \begin{gather}
    \frac12\dfrac{\mathrm d^2{g}}{\mathrm dx^2}(0)
    =k^2r_1r_2+(k+1)^2r_1r_3-r_2r_3
    \label{d2g2}\\
    \frac12\dfrac{\mathrm d^4{g}}{\mathrm dx^4}(0)
    =-k^4r_1r_2-(k+1)^4r_1r_3+r_2r_3
    \label{d2g3}
  \end{gather}
  If~\eqref{d2g2} vanishes, then the sum of~\eqref{d2g3} with
  \eqref{d2g2} yields
  \begin{equation*}
    \frac12\dfrac{\mathrm d^4{g}}{\mathrm dx^4}(0)
    =-k(k+1)r_1\bigl((k-1)kr_2+(k+1)(k+2)r_3\bigr)<0.\qedhere
  \end{equation*}
\end{proof}

\begin{rem}
  We were able to prove directly that the system
  \begin{equation*}
    \left\{
      \begin{aligned}
        &{f(x)}={f(y)}\\
        &\dfrac{\mathrm d{f}}{\mathrm dx}(x)   
        =\dfrac{\mathrm d{f}}{\mathrm dx}(y)=0\\
        &\dfrac{\mathrm d^2{f}}{\mathrm dx^2}(x),   
        \dfrac{\mathrm d^2{f}}{\mathrm dx^2}(y)\le0
      \end{aligned}
    \right.
  \end{equation*}
  implies $x=y$ modulo $2\pi$ or $t=\pi/(k+l)$ and
  $x+y=2m\pi/(k+l)$, but our computations are very involved and
  opaque.
\end{rem}

\begin{rem}
  This proposition is a complex counterpart to \cite[Lemma~2.1.\,ii)]{re95}.  
\end{rem}

\begin{rem}
  Suppose that $l=k=1$. If $t\in\io[0,\pi/2]$, it is necessary to
  solve a generally irreducible quartic equation in order to compute
  the maximum of $f$. If $t=\pi/2$, it suffices to solve a
  linear equation and one gets the following expression for
  \begin{math}
        \max_{x}{\bigabs{r_1\emi x+\iu r_2+r_3\ei x}}:
  \end{math}
  \begin{equation*}
    \begin{cases}
      (r_1+r_3)\sqrt{1+{r_2^2}/{4r_1r_3}}&\text{if $\bigabs{r_1^{-1}-r_3^{-1}}<4r_2^{-1}$}\\
      r_2+\abs{r_3-r_1}&\text{otherwise.}
    \end{cases}
  \end{equation*}
  This formula appears in \cite[(3.1)]{ak01}.  In the first case, the
  maximum is attained at the two points $x^*$ such that $\sin
  x^*=r_2(r_3-r_1)/4r_1r_3$.
\end{rem}

\begin{rem}
  Suppose that $l=1$ and $k=2$. If $t\in\io[0,\pi/3]$, it is
  necessary to solve a generally irreducible sextic equation in order to
  compute the maximum of $f$. If $t=\pi/3$, it suffices to
  solve a quadratic equation and one gets the following expression for
  \begin{math}
    \max_{x}{\bigabs{r_1\emi{2x}+r_2\ei{\pi/3}+r_3\ei x}}:
  \end{math}
  if $r_1^{-1}-4r_3^{-1}<9r_2^{-1}$, then its square makes
  \begin{equation*}
    r_1^2+\frac23r_2^2+r_3^2+r_1r_2+2r_1r_3\biggl[{\biggl({\Bigl(\frac{r_2}{3r_3}\Bigr)}^2+\frac{r_2}{3r_1}+1\biggr)}^{3/2}-{\Bigl(\frac{r_2}{3r_3}\Bigr)}^3\biggr]
  \end{equation*}
  and the maximum is attained at the two points $x^*$ such that
  \begin{equation*}
    2\cos(\pi/3-x^*)={\biggl({\Bigl(\frac{r_2}{3r_3}\Bigr)}^2+\frac{r_2}{3r_1}+1\biggr)}^{1/2}-\frac{r_2}{3r_3};
  \end{equation*}
  otherwise, it makes $-r_1+r_2+r_3$.
\end{rem}



\section{The maximum modulus points of a trigonometric trinomial}
\label{secmm}


\noindent If we undo all the reductions made in Sections~\ref{sec2bis}
and~\ref{sec3} and at the beginning of Section~\ref{sec4}, we get the following theorem.

\begin{thm}
  \label{mmpttt}
  Let $\la_1$, $\la_2$ and $\la_3$ be three pairwise distinct integers such
  that $\la_2$ is between $\la_1$ and $\la_3$. Let $r_1$, $r_2$ and $r_3$ be
  three positive real numbers.  Given three real numbers $t_1$, $t_2$
  and $t_3$, consider the trigonometric trinomial
  \begin{equation*}
    T(x)={{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}+r_3\ei{(t_3+\la_3x)}}}
  \end{equation*}
  for $x\in\R$. Let $d=\gcd(\la_2-\la_1,\allowbreak\la_3-\la_2)$
  and choose integers $a_1$ and $a_3$ such that
  \begin{equation*}
    \tau=\frac{\la_2-\la_3}d(t_1-2\pi a_1)+\frac{\la_3-\la_1}dt_2+\frac{\la_1-\la_2}d(t_3-2\pi a_3),
  \end{equation*}
  satisfies $\abs\tau\le\pi$. Let $\tilde t_1=t_1-2\pi a_1$ and
  $\tilde t_3=t_3-2\pi a_3$.
  \begin{enumerate}
  \item The trigonometric trinomial $T$ attains its maximum modulus at
    a unique point of the interval bounded by
    \begin{math}
      (\tilde t_1-t_2)/\allowbreak(\la_2-\la_1)
    \end{math}
    and
    \begin{math}
      (t_2-\tilde t_3)/\allowbreak(\la_3-\la_2).
    \end{math}
    More precisely,
    \begin{itemize}
    \item if $r_1\abs{\la_2-\la_1}\le r_3\abs{\la_3-\la_2}$, then this point
      is between
      \begin{math}
        (\tilde t_1-\tilde t_3)/(\la_3-\la_1)
      \end{math}
      and
      \begin{math}
        (t_2-\tilde t_3)/(\la_3-\la_2);
      \end{math}
    \item if $r_1\abs{\la_2-\la_1}\ge r_3\abs{\la_3-\la_2}$, then this point
      is between
      \begin{math}
        (\tilde t_1-\tilde t_3)/(\la_3-\la_1)
      \end{math}
      and
      \begin{math}
        (\tilde t_1-t_2)/(\la_2-\la_1);
      \end{math}
    \item $T$ attains its maximum modulus at
      $(\tilde t_1-\tilde t_3)/(\la_3-\la_1)$ if and only if
      $r_1\abs{\la_2-\la_1}\allowbreak=\allowbreak r_3\abs{\la_3-\la_2}$ or
      $\tau=0$.
    \end{itemize}
  \item The function $T$ attains its maximum modulus at a unique point
    modulo $2\pi/d$, and with multiplicity 2, unless $\abs\tau=\pi$.
  \item Suppose that $\abs\tau=\pi$, i.e.,
    \begin{equation}
      \label{opp}
      \frac{\la_2-\la_3}dt_1+\frac{\la_3-\la_1}dt_2+\frac{\la_1-\la_2}dt_3=\pi\mod{2\pi}\text.
    \end{equation}
    Let $s$ be a solution to $2t_1+\la_1s=2t_2+\la_2s=2t_3+\la_3s$ modulo
    $2\pi$: $s$ is unique modulo $2\pi/d$. 
    Then ${T(s-x)}=\ei{(2t_2+\la_2s)}\overline{T(x)}$ for all $x$. Suppose that
    $\abs{\la_3-\la_2}\le\abs{\la_2-\la_1}$. There are three cases.
    \begin{enumerate}
    \item If $\la_2-\la_1=k(\la_3-\la_2)$ with $k\ge2$ 
      integer and
      \begin{equation*}
        r_1^{-1}-k^2r_3^{-1}\ge(k+1)^2r_2^{-1},
      \end{equation*}
      then ${T}$ attains its maximum modulus, $-r_1+r_2+r_3$, only at
      $x=(t_2-\tilde t_3)/\allowbreak(\la_3-\la_2)$ modulo $2\pi/d$,
      with multiplicity 2 if the inequality is strict and with
      multiplicity 4 if there is equality;
    \item if $\la_2-\la_1=\la_3-\la_2$ and
      \begin{equation*}
      \bigabs{r_1^{-1}-r_3^{-1}}\ge4r_2^{-1},
    \end{equation*}
    then ${T}$ attains its maximum modulus, $r_2+\abs{r_3-r_1}$, at a
    unique point $x$ modulo $2\pi/d$, with multiplicity 2 if the
    inequality is strict and with multiplicity 4 if there is
    equality. This point is $(t_2-\tilde
    t_3)/\allowbreak(\la_3-\la_2)$ if $r_1<r_3$, and $(\tilde
    t_1-t_2)/\allowbreak(\la_2-\la_1)$ if $r_3<r_1$;
    \item otherwise ${T}$ attains its maximum modulus at exactly two points
      $x$ and $y$ modulo $2\pi/d$, with multiplicity 2, where $x$ is strictly between
      \begin{math}
        (\tilde t_1-t_2)/\allowbreak(\la_2-\la_1)
      \end{math}
      and
      \begin{math}
        (t_2-\tilde t_3)/\allowbreak(\la_3-\la_2),
      \end{math}
      and $x+y=s$ modulo $2\pi/d$.
    \end{enumerate}
    Note that $s-x=x$ modulo $2\pi/d$ in Cases 1 and 2.
  \end{enumerate}
\end{thm}

\section{\texorpdfstring{Exposed and extreme points of the unit ball
    of $\Cont_\La$}{Exposed and extreme points of the unit ball of
    spaces of trigonometric trinomials}}
\label{sec:exp}


\noindent The characterisation of the maximum modulus points of a trigonometric
trinomial enables us to compute the exposed and the extreme points of the
unit ball of $\Cont_\La$. We begin with a lemma.
\begin{lem}
  \label{deter}
  \begin{enumerate}
  \item A trigonometric trinomial with a given spectrum that attains
    its maximum modulus at two given points modulo $2\pi/d$ is
    determined by its value at these points.
  \item The trigonometric trinomials with a given spectrum that attain their
    maximum modulus with multiplicity 4 at a given point and have a given
    value at this point lie on a parabola.
  \end{enumerate}
\end{lem}


\begin{proof}
  We will use the notation of Theorem~\ref{mmpttt}. Without loss of
  generality, we may suppose that $\la_1=-k$, $\la_2=0$ and $\la_3=l$
  with $k$ and $l$ positive coprime integers. Let $x$ and $y$ be two
  real numbers that are different modulo $2\pi/d$, let $\th$ and
  $\zeta$ be real numbers and let $\rh$ be a positive real number.

  $(a)$. Let us prove that at most one trigonometric trinomial $T$
  attains its maximum modulus at $x$ and $y$ and satisfies
  $T(x)=\rh\ei\th$ and $T(y)=\rh\ei\zeta$. Let us translate $T$ by
  $(x+y)/2$: we may suppose that $x+y=0$. Let us divide $T$ by
  $\ei{(\th+\zeta)/2}$: we may suppose that $\th+\zeta=0$. As $T$
  attains its maximum modulus at the two points $x$ and $y$, we have
  $s=x+y=0$ and $2t_1-ks=2t_2=2t_3+ls=\th+\zeta=0$ modulo $2\pi$.
  Therefore $t_1=t_2=t_3=0$ modulo $\pi$. Let $p_j=\ei{t_j}r_j$: the
  $p_j$ are nonzero real numbers. We have
  \begin{equation*}
    T(x)=p_1\emi{kx}+p_2+p_3\ei{lx}=\rh\ei{\th}\text,
  \end{equation*}
  so that, multiplying by $\emi{\th}$ and taking real and imaginary parts,
  \begin{align}
    p_1\cos(\th+kx)+p_3\cos(\th-lx)&=\rh-p_2\cos\th
    \label{cy}\\
    p_1\sin(\th+kx)+p_3\sin(\th-lx)&=-p_2\sin\th\text.
    \label{sy1}
  \end{align}
  The computation
  \begin{equation*}
    \frac12\frac{\mathrm d{\abs T}^2}{\mathrm dx}(x)
    =\Re\Bigl(\overline{T(x)}\frac{\mathrm dT}{\mathrm dx}(x)\Bigr)
    =\Re\Bigl(\overline{T(x)}\bigl(-\iu k p_1 \emi{kx} + \iu l p_3 \ei{lx}\bigr)\Bigr),
  \end{equation*}
   yields
  \begin{equation}\label{sy2}
    kp_1\sin(\th+kx)-lp_3\sin(\th-lx)=0.
  \end{equation}
  Equations~\eqref{sy1} and~\eqref{sy2} yield $p_1$ and $p_3$ as
  linear functions of $p_2$ because
  $\sin(\th+kx)\allowbreak\sin(\th-lx)\ne0$: otherwise both factors
  would vanish, so that $\th=x=0$ modulo $\pi$ and $x=y$ modulo
  $2\pi$. As $\rh\ne0$, Equation~\eqref{cy} has at most one solution
  in $p_2$.

  $(b)$. We are necessarily in Case~1 or~2 of
  Theorem~\ref{mmpttt}$(c)$, so that we may suppose that $l=1$. Let us
  determine all trigonometric trinomials $T$ that attain their maximum
  modulus at $x$ with multiplicity 4 and satisfy $T(x)=\rh\ei\th$.
  Let us translate $T$ by $x$: we may suppose that $x=0$. Let us
  divide $T$ by $\ei{\th}$: we may suppose that $\th=0$. As $T$
  attains its maximum modulus at 0 with multiplicity 4, we have
  $s-0=0$ and $2t_1-ks=2t_2=2t_3+s=2\th=0$ modulo $2\pi$. Therefore
  $t_1=t_2=t_3=0$ modulo $\pi$.  Let $p_j=\ei{t_j}r_j$: the $p_j$ are
  nonzero real numbers and satisfy the system
  \begin{equation*}
    \left\{
      \begin{aligned}
        &p_1+p_2+p_3=\rh\\
        &k^2p_1p_2+{(k+1)}^2p_1p_3+p_2p_3=0,
      \end{aligned}
    \right.
  \end{equation*}
  that is 
  \begin{equation*}
    \left\{
      \begin{aligned}
        &p_2=\rh-p_1-p_3\\
        &{(kp_1-p_3)}^2=\rh(k^2p_1+p_3).
      \end{aligned}
    \right.
  \end{equation*}
  This is the equation of a parabola.
\end{proof}

\begin{rem}\label{bin:exp}
  The equality
  \begin{equation}
    \label{extr2}
    \max_x\bigabs{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}}=r_1+r_2
  \end{equation}
  shows that the exposed points of the unit ball of the space
  $\Cont_{\{\la_1,\la_2\}}$ are the trigonometric monomials
  $\ei\alpha\e_{\la_1}$ and $\ei\alpha\e_{\la_2}$ with $\alpha\in\R$ and that
  no trigonometric binomial is an extreme point of the unit ball of
  $\Cont_\La$.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{extrpts}~$(a)$]
  \emph{Necessity.}  Let $f\in K$. Let us make four remarks:
  \begin{itemize}
  \item if $f$ is an exposed point, then $\norm f=1$;
  \item if $l$ is a nonzero linear functional on $\Cont_\La$ and $z$ is a
    complex number such that $l^{-1}(z)\cap K=\{f\}$, then
    $\abs{l(f)}=\abs z=\norm l$, so that $l$ attains its norm at $f$;
  \item $f$ is exposed if and only if $\norm f=1$ and there is a
    nonzero linear functional $l$ on $\Cont_\La$ that attains its norm only at multiples of $f$;
  \item if $l$ is a linear functional on $\Cont_\La$ that attains
    its norm at $f$ and $\mu$ is the measure on $[0,2\pi/d\mathclose[$
    identified with the Hahn-Banach extension of $l$ to the space of
    continuous functions on $[0,2\pi/d\mathclose[$, then the support
    of $\mu$ must be a subset of the maximum modulus points of $f$ on
    $[0,2\pi/d\mathclose[$.
  \end{itemize}
  Theorem~\ref{mmpttt}\,$(b,c)$ tells that a trigonometric trinomial
  attains its maximum modulus at one or two points modulo $2\pi/d$. We
  have therefore to show that trigonometric binomials and
  trigonometric trinomials with only one maximum modulus point modulo
  $2\pi/d$ are not exposed.
  \begin{itemize}
  \item A linear functional that attains its modulus at a trigonometric
    binomial attains its norm at a trigonometric monomial because this
    trigonometric binomial is a convex combination of two
    trigonometric monomials with same norm by
    Equation~\eqref{extr2}.
  \item If $f$ attains its maximum modulus at a unique point
    $x\in[0,2\pi/d\mathclose[$, then $l$ must be a multiple of the
    Dirac measure $\delta_x$ at $x$, so that $l$ attains also its norm
    at the monomials in $\Cont_\La$.
  \end{itemize}
  These arguments show also that every linear functional on
  $\Cont_\La$ attains its norm at monomials or at trigonometric
  trinomials with two maximum modulus points.  

  \emph{Sufficiency.}  Conversely, the trigonometric monomial
  $\ei\alpha\e_\la$ is exposed to the linear form
  \begin{equation*}
    P\mapsto\frac1{2\pi}\int_0^{2\pi}P(x)\emi{(\alpha+\la x)}\,\mathrm
  dx.
\end{equation*}
  A trigonometric trinomial $T$ that attains its maximum
  modulus, 1, at two points $x^*_1$ and $x^*_2$ modulo $2\pi/d$ is
  exposed, by Lemma~\ref{deter}\,$(a)$, to any nontrivial convex
  combination of the unimodular multiples of Dirac measures
  \begin{math}
    \overline{T(x^*_1)}\delta_{x^*_1}
  \end{math}
  and
  \begin{math}
    \overline{T(x^*_2)}\delta_{x^*_2}.
  \end{math}
\end{proof}

\begin{rem}
  This is a complex counterpart to Lemma~2.3 in \cite{re95}, dealing with the
  exposed points of the unit ball of the three-dimensional space spanned by the
  functions 1, $\cos x$ and $\cos kx$ in the space of continuous
  functions.
\end{rem}

\begin{proof}[Proof of Theorem~\ref{extrpts}~$(b)$]
  Let $K$ be the unit ball of $\Cont_\La$. Straszewicz's Theorem
  \cite{st35} tells that the exposed points of $K$ are dense in the
  set of its extreme points. Let $P$ be a limit point of exposed
  points of $K$. If $P$ is a trigonometric monomial, $P$ is exposed.
  If $P$ is a trigonometric binomial, $P$ is not an extreme point of
  $K$ by Remark~\ref{bin:exp}. If $P$ is a trigonometric trinomial, it
  is the limit point of trigonometric trinomials that attain their
  maximum modulus twice modulo $2\pi/d$, so that either $P$ also
  attains its maximum modulus twice modulo $2\pi/d$ or, by Rolle's
  Theorem, $P$ attains its maximum modulus with multiplicity 4. Let
  us prove that if a trigonometric trinomial $T$ attains its maximum
  modulus with multiplicity 4 at a point $x$, then $T$ is an extreme
  point of $K$.  Suppose that $T$ is the midpoint of two points $A$
  and $B$ in $K$. Then
  \begin{math}
    \abs{A(x)}\le1,
  \end{math}
  $\abs{B(x)}\le1$ and $\bigl(A(x)+B(x)\bigr)\big/2=T(x)$, so that
  $A(x)=B(x)=T(x)$. Furthermore
  \begin{equation*}
    \abs{T(x+h)}
    \le\frac{\abs{A(x+h)}+\abs{B(x+h)}}2
    =1+\frac{h^2}4\left(\frac{\mathrm d^2{\abs A}}{\mathrm dx^2}(x)+\frac{\mathrm d^2{\abs B}}{\mathrm dx^2}(x)\right)+o(h^2)
  \end{equation*}
  so that, as $\abs{T(x+h)}=1+o(h^3)$,
  \begin{equation*}
    \frac{\mathrm d^2{\abs A}}{\mathrm dx^2}(x)+\frac{\mathrm d^2{\abs B}}{\mathrm dx^2}(x)\ge0~\text{while}~\frac{\mathrm d^2{\abs A}}{\mathrm dx^2}(x),\frac{\mathrm d^2{\abs B}}{\mathrm dx^2}(x)\le0
  \end{equation*}
  and therefore $A$ and $B$ also attain their maximum modulus with
  multiplicity 4 at $x$. As this implies that $A$ and $B$ are
  trigonometric trinomials, Lemma~\ref{deter}\,$(b)$ yields that $T$,
  $A$ and $B$ lie on a parabola: this implies $A=B=T$.
\end{proof}

\begin{rem}
  The set of extreme points of the unit ball of $\Cont_\La$ is not
  closed: for example, if $\la_2$ is between $\la_1$ and $\la_3$,
  every absolutely convex combination of $\e_{\la_1}$ and $\e_{\la_3}$
  is a limit point of exposed points.
\end{rem}

\begin{rem}
  If $\la_2$ is between $\la_1$ and $\la_3$, and $\la_1-\la_2$ is not
  a multiple of $\la_3-\la_2$ nor vice versa, then we obtain that
  every extreme point of the unit ball of $\Cont_\La$ is exposed.
\end{rem}

\begin{rem}
  In particular, compare our description of the extreme points of the
  unit ball of $\Cont_{\{0,1,2\}}$ with the characterisation given by
  K.~M. Dyakonov in \cite[Theorem~1]{dy03}. He shows in his Example~1
  that it is false that ``in order to recognize the extreme points'',
  ``one only needs to know `how often' [the modulus of a trigonometric
  polynomial] takes the extremal value 1''. We show that \emph{with
    the exception of trigonometric binomials}, the extreme points of
  the unit ball of $\Cont_{\{0,1,2\}}$ are characterised by the number
  of zeroes of $1-{\abs{P}}^2$.
\end{rem}

\begin{rem}
  This is a complex counterpart to \cite{ms07}, dealing with the
  extreme points of the unit ball of the three-dimensional space spanned by the
  functions 1, $x^n$ and $x^m$ in the space of real valued continuous
  functions on $[-1,1]$.
\end{rem}


\section{Dependence of the maximum modulus on the arguments}
\label{seccheb}



\noindent We wish to study how the maximum modulus of a trigonometric trinomial
depends on the phase of its coefficients. We shall use the following
formula that gives an expression for the directional derivative of a
maximum function. It was established in \cite{ch43}. Elementary properties of
maximum functions are addressed in \cite[Part Two, Problems 223--226]{ps72}.

\begin{Ch}
  [\protect{\cite[Theorem~VI.3.2, (3.6)]{dm90}}]  
  Let $I\subset\R$ be an open interval and let $K$ be a compact space. Let
  $f(t,x)$ be a function on $I\times K$ that is continuous along
  with $\dfrac{\partial f}{\partial t}(t,x)$. Let
  \begin{equation*}
    f^*(t)=\max_{x\in K}f(t,x).
  \end{equation*}
  Then $f^*(t)$ admits the following expansion at every $t\in I$:
  \begin{equation}
    \label{4}
    f^*(t+h)
    =f^*(t)
    +\max_{f(t,x)=f^*(t)}\Bigl(h\dfrac{\partial f}{\partial t}(t,x)\Bigr)
    +o(h).
  \end{equation}
\end{Ch}

\begin{prp}
  Let $k$ and $l$ be positive coprime integers. Let $r_1$, $r_2$ and
  $r_3$ be three positive real numbers. Then
  \begin{equation*}
    \max_{x}\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}
  \end{equation*}
  is an even $2\pi/(k+l)$-periodic function of $t$ that 
  decreases strictly on $[0,\pi/(k+l)]$: in particular
  \begin{equation*}
    \min_{t}\max_{x}\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}
    =\max_{x}\bigabs{r_1\emi{kx}+r_2\ei{\pi/(k+l)}+r_3\ei{lx}}.
  \end{equation*}
\end{prp}

\begin{proof}
  Let
  \begin{equation}
    \label{f7}
    f(t,x)={\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}}^2.
  \end{equation}
  By~\eqref{pair} and~\eqref{per}, $f^*$ is an even
  $2\pi/(k+l)$-periodic function. 

  Let $t\in\io[0,\pi/(k+l)]$ and choose $x^*$ such that
  $f(t,x^*)=f^*(t)$: then $x^*\in[-t/k,t/l]$ by Proposition~\ref{loc},
  so that
  \begin{equation*}
    \frac1{2r_2}\dfrac{\partial f}{\partial t}(t,x^*)
    =-r_1\sin(t+kx^*)-r_3\sin(t-lx^*)<0
  \end{equation*}
  because $t+kx^*\in[0,(k+l)t/l]$ and $t-lx^*\in[0,(k+l)t/k]$ do
  not vanish simultaneously. By Formula~\eqref{4}, $f^*$ decreases
  strictly on $[0,\pi/(k+l)]$.
\end{proof}

\begin{prp}
  \label{croissant}
  Let $k$ and $l$ be positive coprime integers. Let $r_1$, $r_2$ and
  $r_3$ be three positive real numbers. Then
  \begin{equation}
    \label{gg}
    \frac
    {\max_{x}\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}}
    {\bigabs{r_1+r_2\ei{t}+r_3}}
  \end{equation}
  is an increasing function of $t\in[0,\pi/(k+l)]$. If $kr_1=lr_3$, it is
  constantly equal to 1; otherwise it is strictly increasing. 
\end{prp}

\begin{proof}
  Let $f(t,x)$ be as
  in~\eqref{f7}: then the expression~\eqref{gg} is $g^*(t)^{1/2}$ with
  \begin{gather*}
    g(t,x)=\frac{f(t,x)}{f(t,0)}.
  \end{gather*}
  If $kr_1=lr_3$, then $f(t,0)=f^*(t)$, so that $g^*(t)=1$. As shown in
  the beginning of Section~\ref{sec4}, we may suppose without loss of
  generality that $kr_1<lr_3$.  Let $t\in\io[0,\pi/(k+l)]$ and choose $x^*$
  such that $f(t,x^*)=f^*(t)$: then $x^*\in\io[0,t/l]$ by
  Propositions~\ref{loc} and~\ref{prp-2} and
  \begin{align*}
    \frac{f(t,0)^2}{2r_2}\dfrac{\partial g}{\partial t}(t,x^*)
    &=\frac1{2r_2}\biggl(
    \dfrac{\partial f}{\partial t}(t,x^*)f(t,0)
    -f(t,x^*)\dfrac{\partial f}{\partial t}(t,0)\biggr)\\
      &=\bigl(-r_1\sin(t+kx^*)-r_3\sin(t-lx^*)\bigr)f(t,0)+
      f^*(t)(r_1+r_3)\sin t\\
      &=h(0)f^*(t)-h(x^*)f(t,0)
  \end{align*}
  with 
  \begin{equation*}
    h(x)=r_1\sin(t+kx)+r_3\sin(t-lx).
  \end{equation*}
  Let us show that $h$ is strictly decreasing on
  $[0,t/l]$: in fact, if $x\in\io[0,t/l]$,
  \begin{equation*}
    \frac{\mathrm dh}{\mathrm dx}(x)=kr_1\cos(t+kx)-lr_3\cos(t-lx)
    <(kr_1-lr_3)\cos(t-lx)<0.
  \end{equation*}
  As $f^*(t)>f(t,0)$ and $h(0)>h(x^*)$, $({\partial g}/{\partial
    t})(t,x^*)>0$. By N.~G. Che\-bo\-ta\-r\"ev's formula, $g^*$ increases
  strictly on $[0,\pi/(k+l)]$.
\end{proof}

It is possible to describe the decrease of the maximum modulus of a
trigonometric trinomial independently of the $r$'s as follows.
\begin{prp}
  \label{prp-5}
  Let $k$ and $l$ be two positive coprime integers. Let $r_1$, $r_2$
  and $r_3$ be three positive real numbers. Let $0\le t'<t\le\pi/(k+l)$.
  Then
  \begin{equation}
    \label{ineg-4}
    \max_x\bigabs{r_1\emi{kx}+r_2\ei{t'}+r_3\ei{lx}}
    \le\frac{\cos(t'/2)}{\cos(t/2)}
    \max_x\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}
  \end{equation}
  with equality if and only if $r_1:r_2:r_3=l:k+l:k$.
\end{prp}
\begin{proof}
  Let us apply Proposition~\ref{croissant}. We have
  \begin{align*}
    \frac{{\bigabs{r_1+r_2\ei{t'}+r_3}}^2} {{\bigabs{r_1+r_2\ei{t}+r_3}}^2}
    &=1+\frac{2r_2(r_1+r_3)(\cos t'-\cos t)}{(r_1+r_3)^2+2r_2(r_1+r_3)\cos t+r_2^2}\\
    &=1+\frac{\cos t'-\cos t}{\cos t+\bigl(r_2^2+(r_1+r_3)^2\bigr)/2r_2(r_1+r_3)}\\
    &\le1+\frac{\cos t'-\cos t}{\cos t+1}=\frac{\cos t'+1}{\cos t+1}
  \end{align*}
  by the arithmetic-geometric inequality, with equality if and only if $r_2=r_1+r_3$.
  Therefore Inequality~\eqref{ineg-4} holds, with equality if and only if
  $kr_1=lr_3$ and $r_2=r_1+r_3$.
\end{proof}


We may now find the minimum of the maximum modulus of a trigonometric
trinomial with given spectrum, Fourier coefficient arguments and moduli sum.
Proposition~\ref{prp-5} yields with $t'=0$

\begin{cor}
  \label{prp-3}
  Let $k$ and $l$ be two positive coprime integers. Let $r_1$, $r_2$
  and $r_3$ be three positive real numbers. Let $t\in\iog[0,\pi/(k+l)]$.
  Then
  \begin{equation*}
    \frac{\max_x\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}}{r_1+r_2+r_3}\ge\cos(t/2)
  \end{equation*}
  with equality if and only if $r_1:r_2:r_3=l:k+l:k$.
\end{cor}

\begin{rem}
  There is a shortcut proof to Corollary~\ref{prp-3}:
  \begin{align*}
    \frac{\max_x\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}}{r_1+r_2+r_3}
    &\ge\frac{\bigabs{r_1+r_2\ei{t}+r_3}}{r_1+r_2+r_3}\\
    &=\sqrt{1-\frac{4(r_1+r_3)r_2}{{(r_1+r_2+r_3)}^2}\sin^2(t/2)}\\
    &\ge\sqrt{1-\sin^2(t/2)}=\cos(t/2)
  \end{align*}
  and equality holds if and only if
  $\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}$ is maximal for $x=0$ and
  $r_1+r_3=r_2$.
\end{rem}









\section{\texorpdfstring{\sloppy The norm of unimodular relative Fourier multipliers}
{The norm of unimodular relative Fourier multipliers}}
\label{secnm}


\noindent We may now compute the norm of unimodular relative Fourier multipliers. 

\begin{cor}
  Let $k$ and $l$ be two positive coprime integers. Let
  $t\in[0,\pi/\allowbreak(k+l)]$. Let $M$ be the relative Fourier multiplier
  $(0,t,0)$   that maps the element
  \begin{equation}
    \label{Tkl}
    r_1\ei{u_1}\e_{-k}+r_2\ei{u_2}\e_0+r_3\ei{u_3}\e_{l}
  \end{equation}
  of the normed space $\Cont_{\{-k,0,l\}}$ on
  \begin{equation*}
    r_1\ei{u_1}\e_{-k}+r_2\ei{(t+u_2)}\e_0+r_3\ei{u_3}\e_{l}.
  \end{equation*}
  Then $M$ has norm
  $\cos\bigl(\pi/2(k+l)-t/2\bigr)\big/\cos\bigl(\pi/2(k+l)\bigr)$
  and attains its norm exactly at elements of form~\eqref{Tkl}
  with $r_1:r_2:r_3=l:k+l:k$ and
  \begin{equation*}
    -lu_1+(k+l)u_2-ku_3=\pi\mod{2\pi}.
  \end{equation*}
\end{cor}

\begin{proof}
  This follows from Proposition~\ref{prp-5} and the concavity of
  $\cos$ on $[0,\pi/2]$.
\end{proof}

\begin{rem}
  This corollary enables us to guess how to lift $M$ to an operator that
  acts by convolution with a measure $\mu$. Note that $\mu$ is a
  Hahn-Banach extension of the linear form $f\mapsto Mf(0)$. The
  relative multiplier $M$ is an isometry if and only if $t=0$ and
  $\mu$ is the Dirac measure in 0. Otherwise, $t\ne0$; the proof of
  Theorem~\ref{extrpts}\,$(a)$ in Section~\ref{sec:exp} shows that
  $\mu$ is a linear combination $\alpha\delta_y+\beta\delta_w$ of two
  Dirac measures such that the norm of $M$ is
  $\abs{\alpha}+\abs{\beta}$. Let
  $f(x)=l\emi{kx}+(k+l)\ei{\pi/(k+l)}+k\ei{lx}$: $M$ attains its norm
  at $f$, $f$ attains its maximum modulus at 0 and $2m\pi/(k+l)$,
  and $Mf$ attains its maximum modulus at $2m\pi/(k+l)$, where $m$ is
  the inverse of $l$ modulo $k + l$. As
  \begin{align*}
    (\abs{\alpha}+\abs{\beta})\max_x\abs{f(x)} 
    &=\max_x\abs{Mf(x)}\\
    &=\abs{\mu*f\bigl(2m\pi/(k+l)\bigr)}\\
    &=\abs{\alpha f\bigl(2m\pi/(k+l)-y\bigr)+\beta
      f\bigl(2m\pi/(k+l)-w\bigr)},
  \end{align*}
  we must choose $\{y,w\}=\{0,2m\pi/(k+l)\}$. A computation yields then
  \begin{equation*}
    \mu=\ei{t/2}\frac{\sin\bigl(\pi/(k+l)-t/2\bigr)}{\sin\bigl(\pi/(k+l)\bigr)}\delta_0
    +\ei{(t/2+\pi/(k+l))}\frac{\sin(t/2)}{\sin\bigl(\pi/(k+l)\bigr)}\delta_{2m\pi/(k+l)}.
  \end{equation*}
  If $k=l=1$, this is a special case of the formula appearing in
  \cite[proof of Prop.~1]{hh05}.  Consult \cite{sh61} on this issue.
\end{rem}





\section{The Sidon constant of integer sets}
\label{secsidon}

\noindent Let us study the maximum modulus of a trigonometric trinomial with
given spectrum and Fourier coefficient moduli sum. We get the
following result as an immediate consequence of Corollary~\ref{prp-3}.
\begin{prp}
  \label{prp-4}
  Let $k$ and $l$ be two positive coprime integers. Let $r_1$, $r_2$
  and $r_3$ be three positive real numbers. Let $t\in[0,\pi/(k+l)]$.
  Then
  \begin{equation*}
    \max_x\bigabs{r_1\emi{kx}+r_2\ei{t}+r_3\ei{lx}}
    \ge\cos\bigl(\pi/2(k+l)\bigr)\cdot(r_1+r_2+r_3)
  \end{equation*}
  with equality if and only if $r_1:r_2:r_3=l:k+l:k$ and $t=\pi/(k+l)$.
\end{prp}

This means that the Sidon constant of $\{-k,0,l\}$ equals
$\sec\bigl(\pi/2(k+l)\bigr)$.

The Sidon constant of integer sets was previously known only in the
following three instances:
\begin{itemize}
\item The equality
  \begin{equation*}
    \max_x\bigabs{r_1\ei{(t_1+\la_1x)}+r_2\ei{(t_2+\la_2x)}}=r_1+r_2
  \end{equation*}
  shows that the Sidon constant of sets with one or two elements is 1.
\item The Sidon constant of $\{-1,0,1\}$ is $\sqrt2$ and it is attained for
  $\e_{-1}+2\iu+\e_{1}$. Let us give the original argument: if
  $f(x)=\bigabs{r_1\emi{x}+r_2\ei{t}+r_3\ei{x}}^2$, the parallelogram
  identity and the arithmetic-quadratic inequality yield
  \begin{align*}
    \max_x{f(x)}
    &\ge\max_x\frac{f(x)+f(x+\pi)}2\\
      &=\max_x\frac{
        {\bigabs{r_1\emi{x}+r_3\ei{x}+r_2\ei{t}}}^2+
        {\bigabs{r_1\emi{x}+r_3\ei{x}-r_2\ei{t}}}^2}2\\
      &=\max_x{\bigabs{r_1\emi{x}+r_3\ei{x}}}^2+{\bigabs{r_2\ei{t}}}^2\\
      &=(r_1+r_3)^2+r_2^2\ge\frac{(r_1+r_2+r_3)^2}2.
  \end{align*}
\item The Sidon constant of $\{0,1,2,3,4\}$ is 2 and it is attained for
  $1+2\e_1+2\e_2-2\e_3+\e_4$.
\end{itemize}
These results were obtained by D.~J. Newman (see \cite{sh51}.)  The fact that
the Sidon constant of sets of three integers cannot be 1 had been
noted with pairwise different proofs in \cite{sh51,chm81,li96}.

\begin{rem}
  The real algebraic counterpart is better understood: the maximal
  absolute value of a real algebraic polynomial of degree at most $n$
  with given coefficient absolute value sum is minimal for multiples
  of the $n$th Chebyshev polynomial $T_n$ (look up the last paragraph
  of \cite{er47}, and \cite[Theorem~16.3.3]{rs02} for a proof). As the sum of the absolute values of $T_n$'s
  coefficients is the integer $t_n$ nearest to ${(1+\sqrt{2})}^n/2$, we
  have for real $a_0,a_1,\dots,a_n$
  \begin{equation*}
    \max_{x\in[-1,1]}\bigabs{a_0+a_1x+\dots+a_nx^n}
    \ge t_n^{-1}(\abs{a_0}+\abs{a_1}+\dots+\abs{a_n}).
  \end{equation*}
  \end{rem}



The following estimates for the Sidon constant of large integer sets are known.
\begin{itemize}
\item E.~Beller and D.~J. Newman \cite{bn71} showed that the Sidon
  constant of $\{0,\allowbreak 1,\allowbreak \dots,\allowbreak n\}$ is
  equivalent to $\sqrt n$.
\item (Hadamard sets.) Let $q>1$ and suppose that the sequence
  $(\lambda_j)_{j\ge1}$ grows with geometric ratio $q$:
  $\abs{\la_{j+1}}\ge q\abs{\la_j}$ for every $j$. Then the Sidon
  constant of $\{\la_1,\la_2,\dots\}$ is finite; it is at most $4.27$
  if $q\ge2$ (see \cite{ka07}), at most 2 if $q\ge3$ (see
  \cite{lr75}), and at most $1+\pi^2\big/\left(2q^2-2-\pi^2\right)$ if
  $q>\sqrt{1+\pi^2/2}$ (see \cite[Corollary 9.4]{ne98} or the updated
  \cite[Corollary 10.2.1]{ne99}.)
\end{itemize}

Our computations show that the last estimate of the Sidon constant has the
right order in $q^{-1}$ for geometric progressions.

\begin{prp}
  Let $C$ be the Sidon constant of the geometric progression
  $\{1,\allowbreak q,\allowbreak q^2,\dots\}$, where $q\ge3$ is an
  integer. Then
  \begin{equation*}
    1+\pi^2\big/8(q+1)^2
    \le\sec\bigl(\pi/2(q+1)\bigr)
    \le C\le1+\pi^2\big/\left(2q^2-2-\pi^2\right).
  \end{equation*}
\end{prp}

One initial motivation for this work was to decide whether there are
sets $\{\lambda_j\}_{j\ge1}$ with $\abs{\la_{j+1}}\ge q\abs{\la_j}$
whose Sidon constant is arbitrarily close to 1 and to find evidence
among sets with three elements. That there are such sets, arbitrarily
large albeit finite, may in fact be proved by the method of Riesz
products in \cite[Appendix V, \S1.II]{ks63}; see also
\cite[Proposition 13.1.3]{ne99}. The case of infinite sets remains
open.

A second motivation was to show that the real and complex
unconditional constants of the basis
$(\e_{\la_1},\e_{\la_2},\e_{\la_3})$ of $\Cont_\La$ are different;
we prove however that they coincide, and it remains an open question
whether they may be different for larger sets.  The \emph{real
  unconditional constant} of $(\e_{\la_1},\e_{\la_2},\e_{\la_3})$ is
the maximum of the norm of the eight unimodular relative Fourier
multipliers $(t_1,t_2,t_3)$ such that $t_k=0$ modulo $\pi$. Let
$i,j,k$ be a permutation of $1,2,3$ such that the power of 2 in
$\la_i-\la_k$ and in $\la_j-\la_k$ are equal. Lemma~\ref{isom} shows
that the four relative multipliers satisfying $t_i=t_j$ modulo $2\pi$
are isometries and that the norm of any of the four others, satisfying
$t_i\ne t_j$ modulo $2\pi$, gives the real unconditional constant. In
general, the complex unconditional constant is bounded by $\pi/2$
times the real unconditional constant, as proved in \cite{se97}; in
our case, they are equal.
\begin{cor}
  The complex unconditional constant of the basis
  $(\e_{\la_1},\allowbreak\e_{\la_2},\allowbreak\e_{\la_3})$ of
  $\Cont_\La$ is equal to its real unconditional constant.
\end{cor}

  \begin{acknowledgements}
    I would like to thank Henri Comman, Herv\'e Queff\'elec and
    Harold~S.  Shapiro for their encouragement at crucial moments of this
    research.  The Hungarian-French Scientific and Tech\-no\-lo\-gi\-cal
    Go\-vern\-men\-tal Cooperation, Project \#F-10/04, eased scientific
    contacts that were helpful to this work.
  \end{acknowledgements}

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% \end{thebibliography}

\begin{sloppypar}
  \noindent \textit{Keywords.} Trigonometric trinomial, maximum
    modulus, exposed point, ex\-treme point, Mandel$'$shtam problem, extremal
    problem, relative Fourier multiplier, Sidon constant, unconditional
    constant.  
 \end{sloppypar}
\medskip

\noindent 
\href{http://www.zblmath.fiz-karlsruhe.de/MATH/msc/index}{2000
  \textit{Mathematics Subject Classification.}} Primary 42A05;
Secondary 26D05, 30C10, 42A45, 42A55, 46B15, 46B20.  \medskip

% \noindent
% \href{http://www-math.univ-fcomte.fr/pp_Annu/SNEUWIRTH}{Stefan Neuwirth}, \href{http://www-math.univ-fcomte.fr}{Laboratoire de Math\'ematiques}, Universit\'e de
% Franche-Comt\'e, 25030 Besan\c con cedex, France,
% \href{mailto:stefan.neuwirth@univ-fcomte.fr}{stefan.neuwirth@univ-fcomte.fr}.
% \newpage
% \end{document}


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "h"
%%% End: 
\clearpage

\chapter{\texorpdfstring{On the Sidon constant of $\{0,1,2,3\}$}{On the Sidon constant of \{0,1,2,3\}}}
\label{chquatre}

\begin{abstract}
  \noindent We study an elementary extremal problem on trigonometric polynomials
  of degree $3$. We discover a distinguished torus of extremal
  functions.%% $f_u(t)$ given by
\end{abstract}

\section{Introduction}

\noindent Let $\Lambda=\{\la_0,\la_1,\dots,\la_{n-1}\}$ be a set of $n$
frequencies. We study the following extremal problem: 

\vskip\abovedisplayskip\setbox0=\hbox{$(\dagger)$\quad}\dimen0=\wd0 
\noindent\box0
$\vcenter{
\advance\hsize by -\dimen0
\noindent
Given $n$ positive intensities
$\rh_0, \rh_1, \dots, \rh_{n-1}$, to find $n$ phases
$\th_0,\allowbreak \th_1,\allowbreak \dots,\allowbreak \th_{n-1}$ such that the maximum
$\max_t|\sum\rh_j\ei{\th_j}\ei{\la_jt}|$ is minimal.}$
\vskip\belowdisplayskip

\noindent This should help us to study the
following extremal problem:

\vskip\abovedisplayskip\setbox0=\hbox{$(\ddagger)$\quad}\dimen0=\wd0 
\noindent\box0
$\vcenter{
\advance\hsize by -\dimen0
\noindent
To find $n$ complex coefficients $c_0,c_1,\dots,c_{n-1}$ with given
moduli sum $\abs{c_0}+\abs{c_1}+\dots+\abs{c_{n-1}}=1$
such that the maximum
$\max_t|\sum c_j\ei{\la_jt}|$ is minimal.}$
\vskip\belowdisplayskip 

\noindent Note that this maximum's inverse is the Sidon
constant $S(\Lambda)$. D.~J.~Newman (see \cite[Chapter
3]{sh51}) obtained the following upper bound for $S(\{0,1,\dots,n\})$: by Parseval's theorem on
$\ell^2(\{1,\allowbreak\e^{2\iu\pi/n},\allowbreak\dots,\allowbreak\e^{2\iu(n-1)\pi/n}\})$, putting $\sum
c_j\ei{jt}=f(t)$,
\begin{align}
\max_t|f(t)|^2
&=\max_t|f(t)|^2\vee|f(t+2\pi/n)|^2\vee\dots\vee|f(t+2(n-1)\pi/n)|^2\notag\\
&\ge\max_t\bigl(|f(t)|^2+|f(t+2\pi/n)|^2+\dots+|f(t+2(n-1)\pi/n)|^2\bigr)/n\notag\\
&=\max_t|c_0+c_n\ei{nt}|^2+|c_1|^2+\dots+|c_{n-1}|^2\notag\\
&=\bigl(|c_0|+|c_n|\bigr)^2+|c_1|^2+|c_2|^2+\dots+|c_{n-1}|^2\label{newm}\\
&\ge\bigl(|c_0|+|c_1|+\dots+|c_n|\bigr)^2/n,\notag
\end{align}
and H.~S.~Shapiro showed (ibid.) that equality can
hold exactly if $n\in\{1,2,4\}$. If $n=2$, equality holds
exactly for multiples and translates of $f(t)=1+2\iu\e^{\iu
  t}+\e^{\iu2t}$. If $n=3$, the functions 
\vskip\abovedisplayskip
\centerline{$\displaystyle\frac{\iu2\sqrt2\cos \tau-1-3\sin \tau}{15} + \frac{3+\sin
\tau}{10}\ei t + \frac{3-\sin \tau}{10}\ei{2t} +
\frac{\iu2\sqrt2\cos \tau-1+3\sin \tau}{15}\ei{3t}$}\vskip\belowdisplayskip
\noindent have their modulus bounded by 
$3/5$ for each $\tau$, so that $5/3\le S(\{0,1,2,3\})<\sqrt{3}$.
\vskip\belowdisplayskip
 

The motivation is that we wish to know whether
the real and complex unconditionality constants are distinct for basic
sequences of characters $\ei{nt}$.

\begin{Not}
  $\T=\{\,z\in\C:|z|=1\,\}$ and $\e_\la(z)=z^\la$ for $z\in\T$ and
  $\la\in\Z$.
\end{Not}

\section{\texorpdfstring{Necessary conditions for solutions to Extremal problem $(\dagger)$}{Necessary conditions for solutions to Extremal problem (\textdagger)}}

\noindent Let us the state the theorem of Chebotarev (see
\cite[Th.~VI.3.2, (3.6)]{dm90}) in our context. 

\begin{thm}
Let $\Phi\colon \R^k\times\R\to\R$ be a continuous function
that is $\Cont^1$ in the first variable and periodic
in the second variable. If $\Phi^*\colon x\mapsto\max_y\Phi(x,y)$
achieves its minimum at $x^*$, then there are $r$ points
$y_1,\dots,y_r$ such that $1\le r\le k+1$ and
$\Phi(x^*,y_1)=\dots=\Phi(x^*,y_r)=\Phi^*(x^*)$, and 
$$
\exists\,\alpha_1,\dots,\alpha_r\ge0\quad\hbox{$\sum\alpha_i=1$}\qquad
\sum\alpha_i\frac{\partial\Phi}{\partial x}(x^*,y_i)=0.
$$
\end{thm}

Let us apply this theorem to our problem. For $n$ pairwise distinct integers
$\la_0,\la_1,\dots,\la_{n-1}$, $n$ positive real numbers
$\rh_0,\rh_1,\dots,\rh_{n-1}$ and arguments $t$ and $\th=(\th_0,\th_1,\dots,\th_{n-1})$ let
$$
f(t,\rh,\th)=\frac{\sum\rh_j\ei{(\la_jt+\th_j)}}{\sum\rh_j}=
\Rh(t,\rh,\th)\ei{\Th(t,\rh,\th)}~\text{with}~
\Rh(t,\rh,\th)\ge0~\text{and}~\Phi=\Rh^2.
$$
Define
$$
\Phi^*(\rh,\th)=\max_t\Phi(t,\rh,\th),~
\Phi^*_*=\min\Phi^*$$ 
and, for fixed $\rh_j>0$, 
$$ \Phi_\rh(t,\th) = \Phi(t,\rh,\th),~ \Th_\rh(t,\th) =
\Th(t,\rh,\th),~ \Phi_\rh^*(\th) =
\Phi^*(\rh,\th),~ (\Phi^*_{\rh})_* =
\min\Phi_\rh^*.
$$
Note that $\Phi^*$ is continuous (see \cite[Chapter
5.4]{ps72}). Then 
\begin{eqnarray}
\nonumber\Phi&=&\frac{\sum\sum\rho_j\rho_k\cos((\la_k-\la_j)t+\th_k-\th_j)}
{(\sum\rh_j)^2}\\
\nonumber\frac{\partial\Phi}{\partial t}&=&
\frac{\sum\sum(\la_j-\la_k)\rh_j\rh_k\sin((\la_k-\la_j)t+\th_k-\th_j)}
{(\sum\rh_j)^2}=2\Re\biggl(\bar{f}\frac{\partial f}{\partial t}\biggr)\\
\frac{\partial\Phi}{\partial\th_k}&=&
-2\rh_k\frac{\sum\rh_j\sin((\la_k-\la_j)t+\th_k-\th_j)}{(\sum\rh_j)^2}
=\frac{-2\rh_k}{\sum\rh_j}\Im({\bar f}\ei{(\la_kt+\th_k)})\label{difftheta}
\end{eqnarray}
%
Note that our problem depends in fact on $n-2$ phases only; we have
\begin{equation}
  \label{n-2}
  \sum\frac{\partial\Phi}{\partial\th_k}=0\quad,\quad
\sum\la_k\frac{\partial\Phi}{\partial\th_k}=\frac{\partial\Phi}{\partial t}.
\end{equation}
We get thus
%
\begin{lem}\label{ubc:qlem}
%
Let $\Phi_\rh^*$ achieve its minimum at $\th$ and
$M=\{\,t:\Phi_\rh(t,\th)=(\Phi^*_{\rh})_*\,\}$. 
\begin{enumerate}
\item If $M$ is a singleton $\{t\}$, then $(t,\th)$ is a critical
  point of $\Phi_\rh$. Then all $\la_kt+\th_k$ are congruent to
  $\Th_\rh(t,\th)$ modulo $\pi$.
\item If $M$ is a pair $\{t,u\}$ and $(t,\th),(u,\th)$ are not
  critical points of $\Phi_\rh$, then
$$
\exists\,\mu,\nu>0~\forall\,k\quad
\mu\frac{\partial\Phi}{\partial\th_k}
(t,\th)+\nu\frac{\partial\Phi}{\partial\th_k}(u,\th)=0.
$$
Then $\mu\sin(\Th_\rh(t,\th)-\la_kt-\th_k) +
\nu\sin(\Th_\rh(u,\th)-\la_ku-\th_k)=0$ for all $k$.
\end{enumerate}
\end{lem}

Furthermore one may compute
$$
\frac{\partial\Phi}{\partial\rh_k}=
2\frac{\sum\rh_j\cos((\la_k-\la_j)t+\th_k-\th_j)}{(\sum\rh_j)^2}
-\frac{2\Phi}{\sum\rh_j}
=\frac{2}{\sum\rh_j}\bigl(\Re({\bar f}\ei{(\la_kt+\th_k)})-\Phi\bigr).
$$
%
Note that our problem depends on $n-1$ intensities only; we
have
$$
\sum\rh_k\frac{\partial\Phi}{\partial\rh_k}=0.
$$
Thus
$$\frac{\sum\rh_j}{2}\frac{\partial\Phi}{\partial\rh_k} - 
\iu\frac{\sum\rh_j}{2\rh_k}\frac{\partial\Phi}{\partial\th_k} 
= R(\ei{(\la_kt+\th_k-\Th)}-R). $$
%
\begin{lem}\label{ubc:qlem2}
%
Let $\Phi^*$ achieve its minimum at $(\rh,\th)$ and
$M=\{\,t:\Phi(t,\rh,\th)=\Phi^*_*\,\}$. 

\begin{enumerate}
\item If $M$ is a singleton $\{t\}$, then
  $(t,\rh,\th)$ is a critical point of $\Phi$. Then $\Phi^*_*=1$.
\item If $M$ is a pair $\{t,u\}$ and $\Phi^*_*\ne1$, then 
\begin{equation}\label{eq2max}
\exists\,\mu,\nu>0~\forall\,k\quad
\mu\Bigl(
\frac{\partial\Phi}{\partial\th_k},
\frac{\partial\Phi}{\partial\rh_k}\Bigr)(t,\rh,\th)+
\nu\Bigl(
\frac{\partial\Phi}{\partial\th_k},
\frac{\partial\Phi}{\partial\rh_k}\Bigr)(u,\rh,\th)=0.
\end{equation}
If in turn $\Phi^*_*\ne1$ and \eqref{eq2max} holds, then there
is an $\alpha$ and there are signs $\eps_j\in\{-1,1\}$ not
all equal such that, with
$d_{ijk}=\gcd(\la_k-\la_i,\la_j-\la_i)$ and
$\delta_{ijk}\in\{0,1\}$
$$\left\{\begin{array}{l}
\Rh(t,\rh,\th)=\Rh(u,\rh,\th)=\cos\alpha\\
\sum\eps_i\rh_i=\sum\eps_i\la_i\rh_i=0\\
\forall\, i,j,k\quad\bigl((\la_j-\la_k)\eps_i+(\la_k-\la_i)\eps_j+(\la_i-\la_j)\eps_k\bigr)\alpha
\equiv\\
\hskip13pt(\la_j-\la_k)\th_i+(\la_k-\la_i)\th_j+(\la_i-\la_j)\th_k
\equiv\delta_{ijk}d_{ijk}\pi\mod{2d_{ijk}\pi}\\
\forall\, j~\Th(t,\rh,\th)-\la_jt-\th_j = \eps_j\alpha = 
-(\Th(u,\rh,\th)-\la_ju-\th_j).
\end{array}\right.$$
%
\end{enumerate}
\end{lem}

\begin{proof}
  Let $\Rh=\Rh(t,\rh,\th)=\Rh(u,\rh,\th)$, $\Th_t=\Th(t,\rh,\th)$,
  $\Th_u=\Th(u,\rh,\th)$. Then
$$\forall\,k\quad
\mu(\ei{(\la_kt+\th_k-\Th_t)}-\Rh)+
\nu(\ei{(\la_ku+\th_k-\Th_u)}-\Rh)=0.
$$
Let us suppose $\mu+\nu=1$. By taking moduli, we get then
$$\forall\,k\quad
\cos(\la_kt+\th_k-\Th_t)=\frac{\Rh^2+\mu^2-\nu^2}{2\mu\Rh},~\cos(\la_ku+\th_k-\Th_u)=\frac{\Rh^2+\nu^2-\mu^2}{2\nu\Rh};
$$
Returning to the definition of $f$, we have then
$$
\forall\,k\quad\cos(\la_kt+\th_k-\Th_t)=\Rh=\cos(\la_ku+\th_k-\Th_u),
$$
so that furthermore $\mu=\nu$. Thus we may choose signs
$\eps_k\in\{-1,1\}$ such that
$$\forall\,j,k\quad
\eps_j\sin(\la_jt+\th_j-\Th_t) = \eps_k\sin(\la_kt+\th_k-\Th_t) =
-\eps_k\sin(\la_ku+\th_k-\Th_u).
$$
These sines do not vanish, because otherwise $\Rh=1$. The expressions
of $f$ and $\partial\Phi/\partial t$ yield therefore
$$
\sum\eps_j\rh_j=0,~ \sum\eps_j\la_j\rh_j=0.
$$
%% so that for all $j$ the point $\mu(\ei{(\Th_t-\la_jt+\th_j)}-\Rh)$
%% lies on both circles $(-\mu\Rh,\mu)$ and $(\nu\Rh,\nu)$. As
%% $R\notin\{0,1\}$, they intersect in exactly two conjugate points.
We can therefore choose $\alpha,\alpha'$ and signs $\eps_j\in\{-1,1\}$
such that
$$\forall\, j\qquad\Th_t-\la_jt-\th_j=\eps_j\alpha\quad,\quad 
\Th_u-\la_ju-\th_j=\eps_j\alpha'.$$
$$(\la_k-\la_j)t+\th_k-\th_j=\begin{cases}
  0&\text{if $\eps_j=\eps_k$}\\
  2\eps_j\alpha&\text{otherwise.}
\end{cases}$$ and similarly for $u$.
Finally
\begin{equation*}
  \frac{\la_j-\la_k}{ d_{ijk}}\th_i+\frac{\la_k-\la_i}{
  d_{ijk}}\th_j+\frac{\la_i-\la_j}{ d_{ijk}}\th_k = \pm
\Bigl(\frac{\la_j-\la_k}{ d_{ijk}}\eps_i+\frac{\la_k-\la_i}{
  d_{ijk}}\eps_j+\frac{\la_i-\la_j}{ d_{ijk}}\eps_k\Bigl)\alpha.\qedhere
\end{equation*}
\end{proof}

\section{\texorpdfstring{Necessary conditions for solutions to Extremal problem $(\ddagger)$}{Necessary conditions for solutions to Extremal problem (\textdaggerdbl)}}\label{sec:ext2}

\noindent Let $(\rh,\th)$ solve the extremal problem. If the point $t$
such that $\Phi^*(\rh,\th)=\Phi(t,\rh,\th)$ were unique,
then $\Phi^*(\rh,\th)=1$. If there are exactly two points
$t,u$ such that $\Phi^*(\rh,\th) = \Phi(t,\rh,\th) =
\Phi(u,\rh,\th)$, then either $\Phi^*(\rh,\th)=1$ or there
is an $\alpha$ and there are signs $\eps_0, \eps_1, \eps_2
\in\{-1,1\}$ such that, with $d = \gcd( \la_1-\la_0,
\la_2-\la_0)$,
$$\left\{\begin{array}{l}
\Rh(t,\rh,\th)=\Rh(u,\rh,\th)=\cos\alpha\\
\eps_0\rh_0+\eps_1\rh_1+\eps_2\rh_2=\eps_0\la_0\rh_0+\eps_1\la_1\rh_1+\eps_2\la_2\rh_2=0\\
\bigl((\la_1-\la_2)\eps_0+(\la_2-\la_0)\eps_1+(\la_0-\la_1)\eps_2\bigr)\alpha
\equiv\\
\kern2em(\la_1-\la_2)\th_0+(\la_2-\la_0)\th_1+(\la_0-\la_1)\th_2
\equiv d\pi\mod{2d\pi}%% \\
\end{array}\right.$$
We need not consider the case
$(\la_1-\la_2)\th_0+(\la_2-\la_0)\th_1+(\la_0-\la_1)\th_2\equiv0\mod{2d\pi}$,
for which the system
$$\la_0t+\th_0=\la_1t+\th_1=\la_2t+\th_2$$
has a solution, so that $\Phi^*(\rh,\th)=1$. Then
$$\eps_0=\sgn(\la_1-\la_2)\eps,~\eps_1=\sgn(\la_2-\la_0)\eps,~\eps_2=\sgn(\la_0-\la_1)\eps~\text{for some}~\eps\in\{-1,1\},$$
so that
$$\rh_0=|\la_1-\la_2|\sigma,~ \rh_1=|\la_2-\la_0|\sigma,~
\rh_2=|\la_0-\la_1|\sigma\quad\text{for some }\sigma>0.$$
Choose $\{i,j,k\}=\{0,1,2\}$ such that
$\eps_i\ne\eps_j=\eps_k$. Then $2|\la_j-\la_k|\alpha\equiv
d\pi\mod{2d\pi}$, so that
$R\le\cos(d\pi/2|\la_j-\la_k|)$. Further $\rh_k+\rh_j=\rh_i$, and
the system
$$
\la_jt+\th_j=\la_kt+\th_k=\la_it+\th_i-d\pi/(\la_j-\la_k)
$$
has a solution $t$, for which 
$$R=\frac{|\rh_k+\rh_j+\rh_i\ei{d\pi/(\la_j-\la_k)}|}{2\rh_i}=\cos\frac{d\pi}{2|\la_j-\la_k|}.$$
This solves the problem for $\Lambda=\{0,1,2\}$, as
$t\mapsto\Phi(t,\rh,\th)$ has at most two maxima. 

\section{\texorpdfstring{The case $\{0,1,2,3\}$: a distinguished family of
  polynomials}{The case \{0,1,2,3\}: a distinguished family of
  polynomials}}\label{distpol}

\noindent Let $\Lambda=\{0,1,2,3\}$.  Let $(\rh,\th)$ solve the extremal
problem. If the point $t$ such that $\Phi^*(\rh,\th)=\Phi(t,\rh,\th)$
were unique, then $\Phi^*_*=1$. If there were exactly two points $t,u$
such that $\Phi^*(\rh,\th) = \Phi(t,\rh,\th) = \Phi(u,\rh,\th)$, then
$\Phi^*_*=\sqrt2$. There are therefore exactly three points $t,u,t''$
such that $\Phi^*(\rh,\th) = \Phi(t,\rh,\th) = \Phi(u,\rh,\th)=
\Phi(t'',\rh,\th)$.

Let $f(t,\tau)$ be given by\vskip\abovedisplayskip
\centerline{$\displaystyle\frac{\iu2\sqrt2\cos \tau-1-3\sin \tau}{15} + \frac{3+\sin
\tau}{10}\ei t + \frac{3-\sin \tau}{10}\ei{2t} +
\frac{\iu2\sqrt2\cos \tau-1+3\sin \tau}{15}\ei{3t}.$}\vskip\abovedisplayskip
\noindent One computes that the moduli sum of the coefficients is $1$,
independently of $\tau$. Note that $f(t,-\tau) = \ei{3t}f(-t,\tau)$
and $f(t,\tau+\pi) = \ei{3t}\overline{f(t,\tau)}$, so that we shall
restrict the parameter $\tau$ to $[0,\pi/2]$. Let
$\Phi(t,\tau)=|f(t,\tau)|^2$. We get
\begin{eqnarray*}
\Phi(t,\tau) &=& \frac{2\sqrt2\sin{2\tau}}{75}
(\sin{t}-\sin{2t}+2\sin{3t}) + \frac{247-13\cos{2\tau}}{900} \\
&& {}+
(1+\cos{2\tau}) \Bigl( \frac{\cos{t}}{20} - \frac{\cos{2t}}{25}
\Bigl) + \frac{1+17\cos{2\tau}}{225} \cos{3t}.
\end{eqnarray*}
Let us put 
$$M=\begin{pmatrix}
\displaystyle\frac{2\sin{2t}}{25}-\frac{\sin{t}}{20}-\frac{17\sin{3t}}{75}&
\displaystyle\frac{2\sqrt{2}}{75}(\cos{t}-2\cos{2t}+6\cos{3t})\\
\displaystyle\frac{2\sqrt{2}}{75}(\sin{t}-\sin{2t}+2\sin{3t})&
\displaystyle\frac{13}{900}-\frac{\cos{t}}{20}+\frac{\cos{2t}}{25}-\frac{17\cos{3t}}{225}
\end{pmatrix}
$$
The critical points $(t,\tau)$ of $\Phi$ satisfy 
$$M\begin{pmatrix}\cos{2\tau}\\\sin{2\tau}\end{pmatrix}
=\begin{pmatrix}
  \displaystyle\frac{\sin{t}}{20}-\frac{2\sin{2t}}{25}+\frac{\sin{3t}}{75}\\
  0\vphantom{\frac00}
\end{pmatrix}
$$
We have
$$\det{M} = \frac{1}{6750} \sin{t}
\Bigl(\cos{t}-\frac{1}{4}\Bigr) (4\cos{t}-11)
(16\cos^3{t}-72\cos^2{t}+33\cos{t}-41),$$
which vanishes exactly if
$\cos{t}\in\{-1,1/4,1\}$. Otherwise we get
\begin{equation}\label{CS}
\left\{\begin{array}{rl}
\cos{2\tau} & = \displaystyle- \frac{272\cos^3{t} - 72\cos^2{t} - 159\cos{t} + 23
 }{ 16\cos^3{t}-72\cos^2{t}+33\cos{t}-41}=C(t)\\
\sin{2\tau} & = \displaystyle- \frac{24\sqrt{2} \sin{t} (4\cos{t}+1) (2\cos{t}-1) }{ 16\cos^3{t}-72\cos^2{t}+33\cos{t}-41}=S(t)\end{array}\right.
\end{equation}
Note that this solution is consistent, as $C^2+S^2=1$. For
such $\tau$, $\Phi(t,\tau)=9/25$. Checking the special cases
$\cos{t}\in\{-1,1/4,1\}$ yields that all local maxima are
given by the above formulas, that $\Phi$ attains its global
minimum, $0$, exactly for $\tau=0$ and $t=\pi$, and has exactly one other local
minimum, of value $49/225$, for $\tau=\pi/2$ and $t=0$. There is
exactly one other critical point, of value $5/18$, that
is a saddle point, given by $\tau=\arccos(17/37)/2$, $t=\arccos{1/4}$.

As $C(0)=1$, $C(\pm\pi/3)=-1$, $C(\pm\arccos(-1/4))=1$,
$C(\pi)=-1$, the inter\-mediate values theorem shows that
for a given $\tau$, there are exactly three solutions $t$ to
system \eqref{CS}, for which 
$\Phi(t,\tau)$ achieves then its global maximum, $9/25$.

These formulas yield in turn that for a sign $\eps\in\{-1,+1\}$
\begin{equation}\label{CS2}
\left\{\begin{array}{rl}
\cos{\tau} & = \displaystyle\frac{8\eps(2\cos{t}-1)\cos{t/2}
 }{\sqrt{-16\cos^3{t}+72\cos^2{t}-33\cos{t}+41}}=C_\eps(t)\\
\sin{\tau} & = \displaystyle \frac{3\sqrt{2}\eps(4\cos{t}+1)\sin{t/2}}{\sqrt{-16\cos^3{t}+72\cos^2{t}-33\cos{t}+41}}=S_\eps(t)\end{array}\right.
\end{equation}

\section{\texorpdfstring{The real unconditional constant of $\{0,1,2,3\}$}{The real unconditional constant of \{0,1,2,3\}}}


\noindent If $L$ is a subspace of the space of complex continuous functions on a
compact space~$T$ with $n$~dimensions, then every functional~$l$
on~$L$ extends isometrically to a linear combination of at most
$2n$~Dirac measures: there are $m\le 2n$~points $t_k\in T$ and
coefficients $b_k\in\C$ such that for every $f\in L$ one has
$l(f)=\sum b_kf(t_k)$ and $\|l\|=\sum|b_k|$ (see \cite[Exercice 6.8]{bo81}.) This implies in particular that there is a function $f\in L$ whose maximum modulus points contain the $t_k$.

%On the other hand, given a basis $(f_j)$ of $L$ and any set of $n$~points $t_k$, the $n$ equations $l(f_j)=\sum b_kf_j(t_k)$ are expected to have a unique solution for the coefficients $b_k$, and this would yield an isomorphic extension of $l$. 

Let us now specialise to the case \ $L=\Cont_\Lambda(\T)$ with
$\Lambda$ a finite set. Note that a function in $L$ has at most
$\max\Lambda-\min\Lambda$ maximum modulus points; a trigonometric
trinomial has at most 2 maximum modulus points up to periodicity.

Let us make the ad hoc hypothesis that the~$t_k$ are the $n$th roots
of unity, whose set forms the group $\U_n$: this obliges us to restrict our study to those
functionals~$l$ such that $l(\e_j)=l(\e_{j'})$ if $j\equiv
j'\bmod{n}$. Then the condition~$l(f)=\sum b_kf(t_k)$ reads~
\begin{equation*}
l(\e_j)=\sum_{k=0}^{n-1}b_k\ei{2jk\pi/n}\text{ for $j\in\Lambda$,}
\end{equation*}
which may be
interpreted as telling that the~$l(\e_j)$ are the Fourier coefficients
of the measure $\mu$ on $\U_n$ given by
\begin{equation*}
  \mu=\sum_{k=0}^{n-1}b_k\delta_{\ei{2k\pi/n}}
\end{equation*}
(where the Dirac measures act on $\U_n$). A solution to these equations is given by
\begin{equation*}
  b_k=\frac1n\sum_{j=0}^{n-1}\emi{2jk\pi/n}
  \begin{cases}
    l(\e_{j'})&\text{if there is $j'\equiv j\bmod{n}$ in $\Lambda$}\\
    0&\text{otherwise.}
  \end{cases}
\end{equation*}
The norm of $\mu$ is bounded by
\begin{equation*}
  \sum_{k=0}^{n-1}|b_k|
\end{equation*}
and is attained at $u\in\Cont(\U_n)$ if and only
if~$u(\ei{2k\pi/n})b_k=|b_k|$ for every~$k$, up to a nonzero complex
number. This yields an upper bound for the norm of~$l$ that becomes an
equality if there is an~$f\in\Cont(\T)$ of norm~$1$ such that
$f(\ei{2k\pi/n})=u(\ei{2k\pi/n})$.

Here are two applications.

\begin{prp}
  Let $\Lambda$ be a finite subset of $\Z$. The Sidon constant of
  $\Lambda$ is at most $(\card\Lambda-1)^{1/2}$.
\end{prp}

\begin{proof}
  One may suppose that $\min\Lambda=0$ and choose $n=\max\Lambda$.
  Let~$l$ be a linear functional~$l$ with coefficients~$l(\e_j)$ of
  modulus 1: one may suppose that $l(\e_0)=l(\e_n)$. Then
  \begin{align}
    \|l\|
    &\le\frac1n\sum_{k=0}^{n-1}\biggl|\sum_{j\in\Lambda\setminus\{n\}}l(\e_j)\emi{2jk\pi/n}\biggr|\notag\\
    &\le\Biggl(\frac1n\sum_{k=0}^{n-1}\biggl|\sum_{j\in\Lambda\setminus\{n\}}l(\e_j)\emi{2jk\pi/n}\biggr|^2\Biggr)^{1/2}\label{ytt}\\
    &=\biggl(\sum_{j\in\Lambda\setminus\{n\}}|l(\e_j)|^2\biggr)^{1/2}=(\card\Lambda-1)^{1/2}.\notag\qedhere
  \end{align}
\end{proof}

\begin{rem}
  Herv\'e Queff\'elec showed me a more elementary proof of the same
  fact, that follows the line of D. J.~Newman's computation in the introduction: if $f\in\Cont_\Lambda(\T)$, then there are only $\card\Lambda-1$ squares, and not $n$, in Inequality~\eqref{newm}! 
\end{rem}
\begin{rem}
  If $\Lambda=\{0,1,\dots,n\}$, then Inequality~\eqref{ytt} is an
  equality if and only if $(l(\e_j))_{j=0}^{n-1}$ is a
  \emph{biunimodular} sequence, that is a unimodular function on
  $\U_n$ whose Fourier transform is also unimodular. In other words,
  the matrix $H=(l(\e_{j-k}))_{0\le j,k\le n-1}$ is a \emph{circulant
    complex Hadamard matrix}, where the indices $j-k$ are computed modulo
  $n$: it satisfies $H^*H=n\,\Id$. Such matrices always exist: see
  \cite{bs95}.
\end{rem}

\begin{prp}
\label{cir}
  Let $\Lambda=\{0,1,2,3\}$. The real unconditional constant of $\Cont_\Lambda(\T)$ is $5/3$.
\end{prp}

\begin{proof}
  The polynomials in the previous section show that the real
  unconditional constant of $\Cont_\Lambda(\T)$ is at least $5/3$.
  This constant is the maximum of the norm of linear functionals $l$
  with $l(\e_j)\in\{-1,1\}$. As $l$ has the same norm as $\tilde
  l\colon f\mapsto l(f(\cdot+\pi))$, for which $\tilde
  l(\e_j)=(-1)^jl(\e_j)$, one may suppose that $l(\e_0)=l(\e_3)$. Let
  us now try to lift $l$ to a sum of Dirac measures on the third roots
  of unity. Such a lifting is either the Dirac measure at $0$ or
  \begin{equation*}
    \pm(l(\e_j))_{0\le j\le2}\in\{(-1,1,1),(1,-1,1),(1,1,-1)\}
  \end{equation*}
  and these six cases yield the same norm
  \begin{equation*}
    \frac13(\abs{-1+1+1}+\abs{-1+\ei{2\pi/3}+\ei{4\pi/3}}+\abs{-1+\ei{4\pi/3}+\ei{2\pi/3}})=5/3.\qedhere
  \end{equation*}
\end{proof}

\section[Trigonometric polynomials of degree 3 with real coefficients]
{Trigonometric polynomials of degree 3 with real coefficients}
\noindent Suppose $c_0,c_1,c_2,c_3$ are real and $c_0c_3\ne0$. Let 
$\eps_i=\sgn c_i$ be their sign. Then 
$$\|c_0+c_1\ei{t}+c_2\e^{2\iu t}+c_3\e^{3\iu t}\|_\infty =
|c_0+c_2| + |c_1+c_3|$$ 
if
$$\left|\begin{array}{ll}
&\eps_0=\eps_2\text{ and }\eps_1=\eps_3\\
\text{or}&\left\{\begin{array}{l}
\eps_0\eps_3(c_0c_1+c_1c_2+c_2c_3+9c_0c_3)+4(c_0c_2+c_1c_3)\ge0\\
6|c_0c_3|\le|c_0c_2+c_1c_3|\end{array}\right.\\
\text{or}&
\eps_0\eps_3(c_0c_1+c_1c_2+c_2c_3+9c_0c_3)\le4(c_0c_2+c_1c_3)\\
\text{or}&
\eps_0\eps_1\eps_2\eps_3=-1\text{ and }
(|c_0c_2|+|c_1c_3|)^2\le4(|c_0|+|c_3|)c_0c_3(\eps_0c_1+\eps_3c_2).
\end{array}\right.
$$
Otherwise 
$\|c_0+c_1\ei{t}+c_2\e^{2\iu t}+c_3\e^{3\iu t}\|^2_\infty$ is equal to 
\begin{eqnarray*}
\lefteqn{
2c_0c_3\Biggl(\Bigl(\frac{c_1}{3c_0}+\frac{c_2}{3c_3}\Bigr)^3+\biggl( \Bigl(\frac{c_1}{3c_0}-\frac{c_2}{3c_3}\Bigr)^2 +
\Bigl(1-\frac{c_2}{3c_0}\Bigr) \Bigl(1-\frac{c_1}{3c_3}\Bigr) 
  \biggr)^{3/2} \Biggr)}&&\\
&& \qquad{} + \Bigl(1-\frac{c_1c_2}{3c_0c_3}\Bigr) (c_0^2+c_3^2) - \Bigl(1+\frac{c_1c_2}{ 3c_0c_3}\Bigr)
          (c_0c_2+c_1c_3)  + \frac{2}{3} (c_1^2+c_2^2).
\end{eqnarray*}
Suppose $c_0,c_1,c_2$ are real and $c_0c_2\ne0$. Let 
$\eps_i=\sgn c_i$ be their sign. Then
$$\|c_0+c_1\ei{t}+c_2\e^{2\iu t}\|_\infty = 
|c_0+c_2| + |c_1|\quad\text{if}~\eps_0=\eps_2~\text{or}~ 
|c_1|\,|c_0^{-1}+c_2^{-1}|\ge4;$$
otherwise
$$\|c_0+c_1\ei{t}+c_2\e^{2\iu t}\|^2_\infty = 
(c_0-c_2)^2(1-c_1^2/4c_0c_2).$$


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%\addcontentsline{toc}{chapter}{R\'ef\'erences}
\def\cprime{$'$}\def\bibmath{disponible \`a la biblioth\`eque de
  math\'ematiques}\def\bibirem{disponible \`a la biblioth\`eque de
  l'IREM}\def\BUL{disponible \`a la BU Lettres}\def\BUS{disponible \`a la BU
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\providecommand{\smfedname}{\'ed.}
\providecommand{\smfmastersthesisname}{M\'emoire}
\providecommand{\smfphdthesisname}{Th\`ese}
\begin{thebibliography}{100}

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\end{thebibliography}

\end{document}
