%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % A small sample Â鶹Éçmadou Third Year Project Thesis file. % Any questions to Ian Doust i.doust@unsw.edu.au % % Edited CSG 12.8.2015, make simpler referencing system for theorems etc %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % The first part pulls in a Â鶹Éçmadou Thesis class file. This one is % slightly nonstandard and has been set up to do a couple of % things automatically % \documentclass[ugproject,12pt]{unswthesis} \linespread{1} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{latexsym,amsmath} \usepackage{graphicx} \usepackage{afterpage} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % The following are some simple LaTeX macros to give some % commonly used letters in funny fonts. You may need more or less of % these % \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\F}{\mathbb{F}} \newcommand{\PP}{\mathbb{P}} \newcommand{\T}{\mathbb{T}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\B}{\mathfrak{B}} \newcommand{\BB}{\mathcal{B}} \newcommand{\M}{\mathfrak{M}} \newcommand{\X}{\mathfrak{X}} \newcommand{\Y}{\mathfrak{Y}} \newcommand{\CC}{\mathcal{C}} \newcommand{\E}{\mathbb{E}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cS}{\mathcal{S}} \newcommand{\A}{\mathcal{A}} \newcommand{\ZZ}{\mathcal{Z}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % The following are much more esoteric commands that I have left in % so that this file still processes. Use or delete as you see fit % \newcommand{\bv}[1]{\mbox{BV($#1$)}} \newcommand{\comb}[2]{\left(\!\!\!\begin{array}{c}#1\\#2\end{array}\!\!\!\right) } \newcommand{\Lat}{{\rm Lat}} \newcommand{\var}{\mathop{\rm var}} \newcommand{\Pt}{{\mathcal P}} \def\tr(#1){{\rm trace}(#1)} \def\Exp(#1){{\mathbb E}(#1)} \def\Exps(#1){{\mathbb E}\sparen(#1)} \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} \newcommand{\hatt}[1]{\widehat #1} \newcommand{\modeq}[3]{#1 \equiv #2 \,(\text{mod}\, #3)} \newcommand{\rmod}{\,\mathrm{mod}\,} \newcommand{\p}{\hphantom{+}} \newcommand{\vect}[1]{\mbox{\boldmath $ #1 $}} \newcommand{\reff}[2]{\ref{#1}.\ref{#2}} \newcommand{\psum}[2]{\sum_{#1}^{#2}\!\!\!'\,\,} \newcommand{\bin}[2]{\left( \begin{array}{@{}c@{}} #1 \\ #2 \end{array}\right) } % % Macros - some of these are in plain TeX (gasp!) % \newcommand{\be}{($\beta$)} \newcommand{\eqp}{\mathrel{{=}_p}} \newcommand{\ltp}{\mathrel{{\prec}_p}} \newcommand{\lep}{\mathrel{{\preceq}_p}} \def\brack#1{\left \{ #1 \right \}} \def\bul{$\bullet$\ } \def\cl{{\rm cl}} \let\del=\partial \def\enditem{\par\smallskip\noindent} \def\implies{\Rightarrow} \def\inpr#1,#2{\t \hbox{\langle #1 , #2 \rangle} \t} \def\ip<#1,#2>{\langle #1,#2 \rangle} \def\lp{\ell^p} \def\maxb#1{\max \brack{#1}} \def\minb#1{\min \brack{#1}} \def\mod#1{\left \vert #1 \right \vert} \def\norm#1{\left \Vert #1 \right \Vert} \def\paren(#1){\left( #1 \right)} \def\qed{\hfill \hbox{$\Box$} \smallskip} \def\sbrack#1{\Bigl \{ #1 \Bigr \} } \def\ssbrack#1{ \{ #1 \} } \def\smod#1{\Bigl \vert #1 \Bigr \vert} \def\smmod#1{\bigl \vert #1 \bigr \vert} \def\ssmod#1{\vert #1 \vert} \def\sspmod#1{\vert\, #1 \, \vert} \def\snorm#1{\Bigl \Vert #1 \Bigr \Vert} \def\ssnorm#1{\Vert #1 \Vert} \def\sparen(#1){\Bigl ( #1 \Bigr )} \newcommand\blankpage{% \null \thispagestyle{empty}% \addtocounter{page}{-1}% \newpage} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % These environments allow you to get nice numbered headings % for your Theorems, Definitions etc. % % Environments % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{question}[theorem]{Question} \newtheorem{notation}[theorem]{Notation} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % If you've got some funny special words that LaTeX might not % hyphenate properly, you can give it a helping hand: % \hyphenation{Mar-cin-kie-wicz Rade-macher} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % OK...Now we get to some actual input. The first part sets up % the title etc that will appear on the front page % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{Fractal Analysis in Loop Spaces} \authornameonly{Fred Flintstone} \author{\Authornameonly\\{\bigskip}Supervisor: Professor Karl Marx} \copyrightfalse \figurespagefalse \tablespagefalse %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % And now the document begins % The \beforepreface and \afterpreface commands puts the % contents page etc in % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \beforepreface \afterpage{\blankpage} % plagiarism \prefacesection{Plagiarism statement} \vskip 10pc \noindent I declare that this thesis is my own work, except where acknowledged, and has not been submitted for academic credit elsewhere. \vskip 2pc \noindent I acknowledge that the assessor of this thesis may, for the purpose of assessing it: \begin{itemize} \item Reproduce it and provide a copy to another member of the University; and/or, \item Communicate a copy of it to a plagiarism checking service (which may then retain a copy of it on its database for the purpose of future plagiarism checking). \end{itemize} \vskip 2pc \noindent I certify that I have read and understood the University Rules in respect of Student Academic Misconduct, and am aware of any potential plagiarism penalties which may apply.\vspace{24pt} \vskip 2pc \noindent By signing this declaration I am agreeing to the statements and conditions above. \vskip 2pc \noindent Signed: \rule{7cm}{0.25pt} \hfill Date: \rule{4cm}{0.25pt} \newline \vskip 1pc \afterpage{\blankpage} % Acknowledgements are optional \prefacesection{Acknowledgements} {\bigskip}By far the greatest thanks must go to my supervisor for the guidance, care and support they provided. {\bigskip\noindent}Thanks must also go to Emily, Michelle, John and Alex who helped by proof-reading the document in the final stages of preparation. {\bigskip\noindent}Although I have not lived with them for a number of years, my family also deserve many thanks for their encouragement. {\bigskip\noindent} Thanks go to Robert Taggart for allowing his thesis style to be shamelessly copied. {\bigskip\bigskip\bigskip\noindent} Fred Flintstone, 2 November 2015. \afterpage{\blankpage} % Abstract \prefacesection{Abstract} This thesis is a coherent presentation of a quest to generalise three classical theorems that were discovered in the 1920s, 1930s and 1940s. Their analogues are the product of a conglomeration of ideas that straddle the 1980s and 1990s and the application of these new results brings the story into the twenty-first century. \afterpage{\blankpage} \afterpreface %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Now we can start on the first chapter % Within chapters we have sections, subsections and so forth % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \afterpage{\blankpage} \chapter{Introduction}\label{s-intro} {\noindent}Given a sequence $\{x_n\}_{n=1}^{\infty}$ in a Banach space $\X$, the following natural questions may be asked. \begin{enumerate} \item[(Q1)] Which elements of the space $\X$ can be expressed in the form $\sum_{n=1}^{\infty}a_nx_n$ for some unique sequence $\{a_n\}_{n=1}^{\infty}$ of scalars? \item[(Q2)] Supposing $x\in\X$ can be written as $\sum_{n=1}^{\infty}a_nx_n$, does it matter in what order the terms of the series are summed? \item[(Q3)] In the setting of (Q2), will the series $\sum_{n=1}^{\infty}\varepsilon_na_nx_n$ converge for any choice of $\varepsilon_n=\pm 1$? \item[(Q4)] Given a scalar sequence $\{\phi_n\}_{n=1}^{\infty}$, does it give rise to a bounded linear mapping $\sum_{n=1}^{\infty}a_nx_n\mapsto\sum_{n=1}^{\infty}\phi_na_nx_n$? \end{enumerate} {\noindent}Answering these and similar questions for function spaces whose members act on the circle group $\T\cong[0,2\pi]$ is the main subject of this thesis. In Chapter 1, we give a general picture of the situation, introducing the concepts of Schauder bases and conditional convergence of sequences in Banach spaces. Standard examples show that not every Banach space with a basis $\{x_n\}_{n=1}^{\infty}$ has the property that the terms in each expansion $\sum_{n=1}^{\infty}a_nx_n$ can be freely rearranged. The above questions have been the subject of much study for the trigonometric sequence $\{x_n\}_{n=-\infty}^{\infty}$ contained in $\X=L^p(\T)$ and given by $x_n(t)=e^{int}$. If $1$ for $x^*(x)$. It can be shown that for $1$ for all $x^*\in\X$. The map $x\mapsto\hatt{x}$ from $\X$ into $\X^{**}$ is called the {\em natural map} of $\X$ into its second dual. A Banach space $\X$ is said to be {\em reflexive} if $\X^{**}=\{\hatt{x}:x\in\X\}$. If $\X$ is reflexive then its second dual $\X^{**}$ is isometrically isomorphic to $\X$. However, the converse does not hold (see \cite[III.11]{Con}). It is a consequence of the Riesz Representation Theorem \cite[I.3.4]{Con} that every Hilbert space is reflexive. The spaces $\X=L^p(X,\Omega,\mu)$ and $\lp$ are also reflexive for $1=0$ for all $x^*\in\X^*$. As their names suggest, convergence in the norm topology implies convergence in the weak topology, or in other words, if a sequence converges then it also converges weakly. We shall consider three topologies in the Banach space $\B(\X)$. Suppose $\{T_n\}_{n=1}^{\infty}$ is a sequence of operators in $\B(\X)$. We say it converges to $T$ {\em in norm} if $\norm{T-T_n}\rightarrow 0$, {\em strongly} if $\norm{Tx-T_nx}_{\X}\rightarrow 0$ for all $x\in\X$, and {\em weakly} if $\ip\rightarrow 0$ for all $x\in\X$ and $x^*\in\X^*$. The topologies induced are called the norm operator topology, the strong operator topology and the weak operator topology of $\B(\X)$ respectively. A sequence which converges strongly will converge weakly, and in turn, convergence in the norm topology implies strong convergence. All three topologies have unique limits and distinguish points. The following proposition is easy to verify and is included for familiarisation with the concepts discussed thus far. It shall also be used to establish later results. \begin{proposition}\label{SOT convergence} Let $\X$ be a Banach space and suppose $\{T_n\}_{n=1}^{\infty}$ is a sequence of operators uniformly bounded in norm by some $K>0$ and converging to $T$ in the strong operator topology of $\B(\X)$. Then $\norm{T}\leq K$. \end{proposition} \begin{proof} We want to show $\norm{Tx}_{\X}\leq K\norm{x}_{\X}$ for all $x\in\X$. Fix $x\in\X$ and $\epsilon>0$ and choose an $n\in\N$ such that $\norm{Tx-T_nx}_{\X}<\epsilon$. Then \[\norm{Tx}_{\X}\leq\norm{Tx-T_nx}_{\X}+\norm{T_nx}_{\X} <\epsilon+K\norm{x}_{\X},\] and the proposition follows. \end{proof} %%%%%%%%%%%% \section{Bases in Banach spaces}\label{bases} %%%%%%%%%%%% The definition and usefulness of a basis in a finite dimensional vector space is well-known. It is natural then to want an analogous concept for Banach spaces. The most useful approach is found in the notion of a Schauder basis. A comprehensive introduction to such bases is \cite[pp. 1--52]{Lind}, upon which most of the material in this section is based. \begin{definition} A sequence $\{x_n\}_{n=1}^{\infty}$ in a Banach space $\X$ is called a {\em Schauder basis of $\X$} if, for every $x\in\X$, there is a unique sequence of scalars $\{a_n\}_{n=1}^{\infty}$ such that $x=\sum_{n=1}^{\infty}a_nx_n$. A sequence $\{x_n\}_{n=1}^{\infty}$ which is a Schauder basis for its closed linear span is called a {\em basic sequence}. \end{definition} A Banach space $\X$ is called {\em separable} if it has a countable dense subset. Thus if $\X$ has a Schauder basis then $\X$ is separable. In general the converse is not true. Since the class of Schauder bases is the only type of bases considered in this thesis, it will henceforth be implicit that any discussion about bases refers only to Schauder bases. \begin{example} The ordered set of unit vectors $e_n=(0,0,0,\ldots,1,0,\ldots)$, where the $1$ occurs in the $n$th coordinate, forms a basis in each of the spaces $c$, $c_0$ and $\lp$. Another example of a basis in $c$ is given by $x_1=(1,1,1,\ldots)$ and $x_n=e_{n-1}$ for $n>1$. If $x=(b_1,b_2,\ldots)\in c$, then the expansion of $x$ with respect to this basis is \[x=(\lim_{k\rightarrow\infty}a_k)x_1+(a_1-\lim_{k\rightarrow\infty}a_k)x_2+ (a_2-\lim_{k\rightarrow\infty}a_k)x_3+\cdots.\] \end{example} If a Banach space $\X$ has a basis, we may consider $\X$ as a sequence space via an identification of each element $x=\sum_{n=1}^{\infty}a_nx_n$ in $\X$ with the unique sequence of coefficients $(a_1,a_2,a_3,\ldots)$. To do this note that it is essential to describe the basis as an ordered sequence rather than merely as a set. The following example highlights a deeper reason for why we must specify the ordering of the vectors in a basis. \begin{example}\label{summing basis} Consider the sequence of vectors $\{x_n\}_{n=1}^{\infty}$ in $c$ given by \[x_n=(\underbrace{0,0,\ldots,0}_{n-1},1,1,\cdots),\] for $n\in\Z^+$. It is not hard to see that this is a basis for $c$. It is called the {\em summing basis} in $c$. Consider the sequence $b$ defined by \[b_n=\sum_{k=1}^n(-1)^{k+1}\frac{1}{k}\] for $n\in\Z^+$. By Leibniz' test for alternating series, $b_n$ has a limit as $n\rightarrow\infty$, so $b\in c$. The expansion of $b$ with respect to the summing basis is $b=\sum_{n=1}^{\infty}a_nx_n$ where $a_n=(-1)^{n+1}\frac{1}{n}$. Now consider the following rearrangement of the series. Define $\pi:\Z^+\rightarrow\Z^+$ by \[\pi(n)=\left\{\begin{array}{ll} 2k & \mbox{if $n=3k$}\\ 4k+1 & \mbox{if $n=3k+1$}\\ 4k+3 & \mbox{if $n=3k+2$.} \end{array}\right. \] Then $\pi$ is clearly a permutation of $\Z^+$. However, the expansion \[b'\equiv\sum_{n=1}^{\infty}a_{\pi(n)}x_{\pi(n)}\] does not lie in $c$ since \[b_{3n}'=\sum_{k=1}^n\left(\frac{1}{2k-1}+\frac{1}{2k+1}-\frac{1}{2k}\right) \geq 2\sum_{k=1}^n\frac{1}{k}\] diverges as $n\rightarrow\infty$. \end{example} Not every basis has the defect that the convergence of an expansion with respect to the basis is dependent on the order of summation of the expansion. For example, expansions in finite dimensional spaces can be summed in any order without altering convergence. The following proposition gives various methods for checking whether a sum of vectors can be freely rearranged while still respecting convergence. \begin{proposition}\label{unconditional conv} \cite[Proposition 1.c.1]{Lind} Let $\{x_n\}_{n=1}^{\infty}$ be a sequence of vectors in a Banach space $\X$. Then the following are equivalent. (i) The series $\sum_{n=1}^{\infty}x_{\pi(n)}$ converges for every permutation $\pi$ of the integers. (ii) The series $\sum_{k=1}^{\infty}x_{n_k}$ converges for every choice of $n_10$ there exists an integer $n_0$ such that $\ssnorm{\sum_{j\in J}x_j}_{\X}<\epsilon$ for every finite set of integers $J$ which satisfies $\min\{j\in J\}>n_0$. \end{proposition} With the aid of the proposition, there are now easier ways to verify that the series $\sum_{n=1}^{\infty}a_nx_n$ given in Example \reff{bases}{summing basis} is dependent on the order of summation. Simply observe that $\sum_{n=1}^{\infty}a_{2n-1}x_{2n-1}$ does not converge in $c$ and hence statement (ii) of the proposition fails to hold. \begin{definition} Let $\{x_n\}_{n=1}^{\infty}$ be a sequence of vectors in a Banach space $\X$. A series $\sum_{n=1}^{\infty}x_n$ which satisfies any of the conditions (i), (ii), (iii) or (iv) in Proposition \reff{bases}{unconditional conv} is said to be {\em unconditionally convergent}. A basis $\{x_n\}_{n=1}^{\infty}$ of a Banach space $\X$ is {\em unconditional} if for every $x\in\X$, its expansion $x=\sum_{n=1}^{\infty}a_nx_n$ converges unconditionally. Otherwise the basis is said to be {\em conditional}. \end{definition} Thus the standard ordered bases $\{e_n\}_{n=1}^{\infty}$ for $c$, $c_0$ and $\lp$, $1\leq p<\infty$ are unconditional bases. This means that any reordering of $\{e_n\}_{n=1}^{\infty}$ also yields unconditional bases in these spaces. The summing basis for $c$ is clearly a conditional basis. Given a basis $\{x_n\}_{n=1}^{\infty}$, one might ask what complex sequences $\{\phi_n\}_{n=1}^{\infty}$ give rise to a bounded multiplier transform? That is, is there a constant $C>0$ such that for all $x\in\X$, \[\snorm{\sum_{n=1}^{\infty}\phi_na_{n,x}x_n}_{\X}\leq C\norm{x}_{\X},\] where $\sum_{n=1}^{\infty}a_{n,x}x_n$ is the expansion of each $x\in\X$ with respect to the basis? If that basis is unconditional, then the space of such sequences is just $\ell^{\infty}$ (see \cite[Proposition 1.c.7]{Lind}). We have seen that conditional bases do exist, and some of the resulting difficulties that arise when working with a conditional basis. Is there any way in which we can avoid such problems in separable Banach spaces? In other words, can we always find an unconditional basis for a separable Banach space? The answer is no. The space $L^1[0,1]$ has no unconditional basis. In fact, $L^1[0,1]$ cannot even be embedded in a space with an unconditional basis \cite[Propostion 1.d.1]{Lind}. There are many open problems regarding unconditional bases. For example, suppose $\X$ is a Banach space with an unconditional basis, and let $\Y$ be a complemented subspace of $\X$. Does $\Y$ have an unconditional basis? One outstanding problem has recently been solved (see \cite{Gowers}). Does every infinite dimensional Banach space $\X$ contain an unconditional basic sequence? The answer is no. We shall not pursue such broad problems in this thesis. Instead, we shall use tools from harmonic analysis and spectral theory to examine whether or not particular sequences in certain Banach spaces are conditional, and whether or not they form bases for those spaces. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter{Classical Harmonic Analysis}\label{cha} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We now leave the broad discussion about bases and conditionality in Banach spaces and instead focus our attention to some particular classes of the $L^p$ Banach spaces. The theory of classical harmonic analysis provides a good starting point from which to tackle problems relating to bases and conditionality in this setting. In this chapter we give an overview of some basic results from classical harmonic analysis and Fourier theory. We begin by introducing the fundamental concepts of harmonic analysis on locally compact Abelian groups. The Fourier transform and Fourier multipliers will be defined. Connections between multiplier transforms and convolution operators will be drawn in Section \ref{lca}, as well as important examples given. In Section \ref{lca} we discuss some of the classical results of M. Riesz, Littlewood--Paley and Marcinkiewicz. The general theory given in this chapter can be used to consider the the set of functions $\{\varphi_n\}_{n\in\Z}\subset L^p(\T)$, where $\varphi_n:t\mapsto e^{int}$. Is this a basis for the Banach space $L^p(\T)$? Is it an unconditional basic sequence? If not, how close is it to being an unconditional sequence? The results stated in Section \ref{lca} answer such questions. %%%%%%%%%%%%%%%% \section{Harmonic Analysis on Locally Compact Abelian Groups}\label{lca} %%%%%%%%%%%%%%%% In this section we give the necessary background from which we can begin to answer the questions from above. The theory mentioned here can be found in \cite{Katznelson} and is quite standard. In what follows, if $G$ is a group, the inverse of an element $x\in G$ will be denoted by $-x$, and the group operation from $G\times G$ into $G$ will be denoted by $(x,y)\mapsto x+y$. \begin{definition} A {\em locally compact Abelian group} (or an LCA group) $G$ is an Abelian group which is also a locally compact Hausdorff space such that the group operations $x\mapsto -x$ from $G$ onto $G$ and $(x,y)\mapsto x+y$ from $G\times G$ onto $G$ are continuous. \end{definition} The most-studied examples of LCA groups are the integers $\Z$, the circle group $\T$ and real line $\R$ with their usual topologies. It is easy to verify that any Abelian group can be made into an LCA group when endowed with the discrete topology. In this thesis the circle group $\T$ will feature most often. It can be modelled by the unit circle $\{\omega\in\C:|\omega|=1\}$ in the complex plane, or as the quotient group $\R/2\pi\Z$. We shall usually adopt the latter model, thinking of $\T$ as the interval $[0,2\pi]$ with endpoints identified and addition as the group operation. A good reason to study LCA groups is that we can use them to construct measure spaces that are translation invariant. \begin{definition} Let $G$ be a locally compact Abelian group. A {\em Haar measure} on $G$ is a positive regular Borel measure $\mu$ having the following two properties: (i) $\mu(E)<\infty$ if $E\subseteq G$ is compact; and (ii) $\mu(E+x)=\mu(E)$ for all measurable $E\subseteq G$ and all $x\in G$. \end{definition} \begin{theorem}\cite[Chapter VII, \S 2]{Katznelson} Let $G$ be an LCA group. Then a Haar measure on $G$ exists and is unique up to multiplication of a positive constant. \end{theorem} Hence one often speaks of {\em the} Haar measure. For $G=\T$, Haar measure is usually taken to be normalised Lebesgue measure, $(2\pi)^{-1}dt$. If $G$ is discrete and infinite, Haar measure is usually normalised to have mass one at each point. We denote the $L^p$ space of functions on $G$ with respect to Haar measure by $L^p(G)$. As mentioned in Example \reff{Banach Spaces}{L^p example}, we will not usually distinguish between functions defined on $G$ and their $L^p$ equivalence classes. Using Haar measure, we may turn $L^1(G)$ into a Banach algebra. \begin{theorem}\label{convolution} Let $G$ be an LCA group with Haar measure $dy$ and suppose $f,g\in L^1(G)$. Then for almost all $x\in G$, the function $y\mapsto f(x-y)g(y)$ for $y\in G$ is integrable on $G$. If we write \[h(x)=\int_G f(x-y)g(y)dy,\] then $h\in L^1(G)$ and $\norm{h}_1\leq\norm{f}_1\norm{g}_1$. \end{theorem} \begin{definition} Let $G,f,g$ be as in Theorem \reff{lca}{convolution}. Then the {\em convolution} of $f$ and $g$, denoted $f*g$, is given by \[(f*g)(x)=\int_G f(x-y)g(y)dy\] for almost all $x\in G$ \end{definition} \begin{corollary} Let $G$ be an LCA group. Then $L^1(G)$ is a Banach Algebra under convolution and pointwise addition. \end{corollary} Our next aim is to define the Fourier transform of a function in $L^1(G)$. We start by introducing the characters of $G$. \begin{definition} A {\em character} on an LCA group $G$ is a continuous mapping $\xi$ from $G$ into $\C$ such that $|\xi(x)|=1$ and $\xi(x+y)=\xi(x)\xi(y)$ for all $x,y\in G$. The set of all characters of $G$ is denoted $\hatt{G}$. \end{definition} Thus a character is a continuous homomorphism of $G$ into $\T$. It can be shown that the set of characters on $\T$ is the set $\{\varphi_n\}_{n\in\Z}$, where $\varphi_n(t)=e^{int}$ for all $n\in\Z$ and all $t\in[0,2\pi]$. Every character of $\Z$ has the form $n\mapsto e^{int}$ for some $t\in[0,2\pi]$. The characters of $\R$ are all of the form $x\mapsto e^{ixy}$ for some $y\in\R$. For any LCA group $G$, the set of characters $\hatt{G}$ can be made an Abelian group. We shall denote its group operation by $+$ (in what follows, this should cause no confusion with the group operation $+$ for $G$) and define it by $(\xi_1+\xi_2)(x)=\xi_1(x)\xi_2(x)$ for all $x\in G$. We write $\ip$ for $\xi(x)$. A topology is defined on $\hatt{G}$ by specifying that $\{\xi_n\}_{n=1}^{\infty}\subset\hatt{G}$ converges to $\xi\in\hatt{G}$ if $\{\xi_n\}_{n=1}^{\infty}\subset\hatt{G}$ converges uniformly to $\xi$ on all compact subsets of $G$. It can be shown that this topology turns $\hatt{G}$ into a locally compact Abelian group (see \cite[Chapter VII, \S 3]{Katznelson}). We say that $\hatt{G}$ is the {\em dual group} of $G$. The {\em Pontryagin Duality Theorem} (see \cite[p.189]{Katznelson}) asserts that, for $x\in G$ and $\xi\in\hatt{G}$, the mapping $\xi\mapsto \ip$ is a character on $\hatt{G}$, and every character on $\hatt{G}$ is of this form. Moreover, the topology induced by uniform convergence on compact subsets of $\hatt{G}$ coincides with the original topology on $G$. In otherwords, if $\hatt{G}$ is the dual group of $G$, then $G$ is the dual of $\hatt{G}$. We illustrate the Pontryagin Duality Theorem for the LCA group $\T$. We saw that there exists a bijection between $\hatt{\T}$ and $\Z$, since given any $\xi\in\hatt{\T}$, there is a unique $n\in\Z$ such that $\ip=e^{int}$ for all $t\in\T$. Moreover, the topology $\hatt{T}$ is the discrete topology. Thus $\hatt{\T}\simeq\Z$. One may similarly show that $\hatt{\Z}\simeq\T$. This duality may be described (by abuse of notation) as $\ip=e^{int}=\ip$ for all $t\in\T$ and all $n\in\Z$. (On the left hand side, we regard $n$ as an element of $\Z$ while on the right hand side we regard it as a function on $\T$.) We now have all the tools needed to define the Fourier transform of an integrable function on an LCA group. \begin{definition}\label{Fourier transf} Let $G$ be an LCA group. Then the {\em Fourier transform} of a function $f\in L^1(G)$ is defined by \[\hatt{f}(\xi) = \int_G \overline{\ip}f(y)dy\] for all $\xi\in\hatt{G}$, where $dy$ is Haar measure on $G$. \end{definition} Thus the Fourier transform of a function $f\in L^1(\Z)$ is \[\hatt{f}(t)=\sum_{n=-\infty}^{\infty}e^{-int}f(n)\] for all $t\in\T$, since Haar measure on $\Z$ is unit point-mass measure. Similarly, the Fourier transform of a function $f\in L^1(\T)$ is \[\hatt{f}(n) = \frac{1}{2\pi}\int_0^{2\pi}e^{-int}f(t)dt\] for all $n\in\Z$, where $dt$ is Lebesgue measure. The {\em Fourier series} of $f$ is the corresponding formal expression \[\sum_{n=-\infty}^{\infty}\hatt{f}(n)\varphi_n\] where $\varphi_n(t)=e^{int}$ for all $t\in\T$ and all $n\in\Z$. We aim to use the Fourier transform to construct a large class of operators which act on the $L^p$ space of a given LCA group. We may do this via {\em Plancherel's Theorem}. \begin{theorem}\label{isometry}\cite[Chapter VII, \S 4]{Katznelson} {\em Plancherel's Theorem.} Let $G$ be a locally compact Abelian group. Then the Fourier transform on $L^1(G)$ is an isometry of $L^1\cap L^2(G)$ onto a dense subspace of $L^2(\hatt{G})$. Hence it can be extended to an isometry of $L^2(G)$ onto $L^2(\hatt{G})$. \end{theorem} For the circle group $\T$, Plancherel's theorem takes the following form. \begin{theorem}\label{Plancherel for T}\cite[Theorem I.5.5]{Katznelson} (i) Let $f\in L^2(\T)$. Then \[\sum_{n=\infty}^{\infty}|\hatt{f}(n)|^2= \frac{1}{2\pi}\int_0^{2\pi}|f(t)|^2\,dt,\] that is, $\ssnorm{\hatt{f}}_{L^2(\Z)}=\norm{f}_{L^2(\T)}$. (ii) For each $f\in L^2(\T)$, \[f=\lim_{N\rightarrow\infty}\sum_{n=-N}^N\hatt{f}_n\varphi_n\] in $L^2(\T)$ norm, where $\varphi_n:t\mapsto e^{int}$. (iii) Given any sequence $\{a_n\}_{n=-\infty}^{\infty}$ of complex numbers in $L^2(\Z)$, then there exists a unique $f\in L^2(\T)$ such that $a_n=\hatt{f}(n)$. Thus the correspondence $f\leftrightarrow\{\hatt{f}(n)\}_{n=-\infty}^{\infty}$ is an isometry between $L^2(\T)$ and $L^2(\Z)$. \end{theorem} A by-product of the above theorem is that the set $\{\varphi_n\}_{n\in\Z}$ of functions in the Hilbert space $L^2(\T)$ forms a complete orthonormal system (and hence an unconditional basis) with respect to the inner product \[\ip=\frac{1}{2\pi}\int_0^{2\pi}f(t)\overline{g(t)}\,dt.\] It is harder to establish whether or not $\{\varphi_n\}_{n\in\Z}$ is a basis for $L^p(\T)$ when $1\leq p<\infty$ and $p\neq2$. Standard results about Fej\'{e}r's kernel (see Section \ref{lca} and \cite[I.2.6]{Katznelson}) give the following facts. First recall that a {\em trigonometric polynomial} is a function on $\T$ of the form $\sum_{n=-N}^Na_n\varphi_n$ where $N\in\N$ and $a_n\in\C$. \begin{theorem}\label{Fejer facts} Fix $1\leq p<\infty$. Then (i) the trigonometric polynomials are dense in $L^p(\T)$; (ii) if $f,g\in L^p(\T)$ and $\hatt{f}(n)=\hatt{g}(n)$ for all $n\in\Z$, then $f=g$; and (iii) if the Fourier series of a function $f\in L^p(\T)$ does converge in $L^p$-norm, then it converges to $f$. \end{theorem} So if the Fourier series does converge for each function in $L^p(\T)$, $\{\varphi_n\}_{n\in\Z}$ is a basis for $L^p(\T)$. Otherwise, one would like to know whether or not $\{\varphi_n\}_{n\in\Z}$ is a basic sequence in $L^p(\T)$ for $p\neq2$, and whether this sequence is unconditional. One way to study such problems is through multiplier transforms. \begin{definition}\label{multipliers} Let $G$ be an LCA group and let $\phi:\hatt{G}\rightarrow\C$ be bounded and measurable on $\hatt{G}$. By Plancherel's Theorem, define $S_{\phi}$ to be the continuous linear mapping of $L^2(G)$ into itself for which $({S_{\phi}f})\,\hatt{\,} = \phi \hatt{f}$ for all $f\in L^2(G)$. Then for $1\leq p\leq\infty$, $\phi$ is said to be an {\em $L^p(G)$ multiplier} if and only if there is a constant $C>0$ such that \begin{equation}\label{eq-multiplier norm} \norm{S_{\phi}f}_p\leq C_p\norm{f}_p \end{equation} for all $f\in L^2\cap L^p(\T)$. In this case $S_{\phi}$ is called the {\em multiplier transform corresponding to $\phi$ on $L^p(G)$} and $\phi$ may also be referred to as a {\em Fourier multiplier} or a {\em multiplier function} for $L^p(G)$. \end{definition} The space of Fourier multipliers for $L^p(G)$ will be denoted by $M_p(\hatt{G})$. We turn $M_p(\hatt{G})$ into a Banach space by equipping it with the following norm: for $\phi\in M_p(\hatt{G})$ define $\norm{\phi}_{M_p(\hatt{G})}$ to be the usual operator norm on $L^p(G)$ for the operator $S_{\phi}$. Thus $\norm{\phi}_{M_p(\hatt{G})}$ is the smallest possible $C\geq 0$ for which (\ref{eq-multiplier norm}) holds. It is trivial to check from the definitions that if $\phi_1,\phi_2\in M_p(\hatt{G})$, then the product $\phi_1\phi_2$, defined by pointwise operations, is in $M_p(\hatt{G})$ and \[\norm{\phi_1\phi_2}_{M_p(\hatt{G})}\leq\norm{\phi_1}_{M_p(\hatt{G})} \norm{\phi_2}_{M_p(\hatt{G})}.\] In fact, we have the following result. \begin{proposition}\label{multiplier algebra}\cite[\S3]{BG Spectral} Let $G$ be an LCA group. Then for $1\leq p<\infty$, $M_p(\hatt{G})$ is a Banach algebra under pointwise operations, and the mapping $\phi\mapsto S_{\phi}$ is an identity-preserving algebra homomorphism of $M_p(\hatt{G})$ into $\B(L^p(G))$. \end{proposition} The next example illustrates some of the concepts that have been introduced so far. \begin{example}\label{lca example} Let $G=\T$ and let $\varphi_n:t\mapsto e^{int}$ for all $t\in\T$ and $n\in\Z$. Fix an $m\in\Z$ and consider the characteristic function $\chi_{\{m\}}$ on $\Z$. Then for any $f\in L^2(\T)$ we have, by Definition \reff{lca}{multipliers}, \[\bigl(S_{\chi_{\{m\}}}f\bigr)\,\hatt{\,}\,(n)=\chi_{\{m\}}(n)\hatt{f}(n) =\chi_{\{m\}}(n)\hatt{f}(m)=\hatt{\varphi}_m(n)\hatt{f}= \bigl(\hatt{f}(m)\varphi_m\bigr)\,\hatt{\,}\,(n)\] for all $n\in\Z$. Thus $S_{\chi_{\{m\}}}f=\hatt{f}(m)\varphi_m$, and $S_{\chi_{\{m\}}}$ projects each $f$ onto the $m$th term of its Fourier series. Now we consider the convolution product of $f$ with $\varphi_m$. For almost all $t\in\T$ we have \begin{eqnarray*} (\varphi_m*f)(t) &=& \frac{1}{2\pi}\int_0^{2\pi}\varphi_m(t-s)f(s)ds \\ & = & e^{imt}\,\frac{1}{2\pi}\int_0^{2\pi}e^{-ims}f(s)ds \\ & = & e^{imt}\hatt{f}(m). \end{eqnarray*} Thus $S_{\chi_{\{m\}}}$ and the operator $f\mapsto\varphi_m*f$ coincide on $L^2(\T)$. It is easy to see that $\ssnorm{S_{\chi_{\{m\}}}}=1$ and hence $\ssnorm{\chi_{\{m\}}}_{M_p(\hatt{G})}=1$. \end{example} We now give one reason why multiplier transforms are of interest to us. Let $\varphi_n(t)=e^{int}$ for each $n\in\Z$ and $t\in\T$. One might hope that $\{\varphi_n\}_{n\in\Z}$ is a basis for $L^p(\T)$. Assume for the moment that this is the case. The expansion of a function $f\in L^p(\T)$ with respect to this basis is then given by $\sum_{n\in\Z}\hatt{f}(n)\varphi_n$. If one wanted to prove that the basis was unconditional, it would suffice to find (by Proposition \reff{bases}{unconditional conv}) a constant $C_p$ depending only on $p$ such that \[\snorm{\sum_{n\in\Z}\varepsilon_n\hatt{f}(n)\varphi_n}_p\leq C_p\norm{f}_p\] for all $f\in L^p(\T)$ and all choices $\varepsilon_n=\pm 1$. This would be equivalent to finding a constant $C_p$ such that \[\norm{S_{\varepsilon}f}_p\leq C_p\norm{f}_p\] for all $f\in L^2\cap L^p(\T)$ and all functions $\varepsilon:\Z\rightarrow\C$ taking values in $\{-1,1\}$. Motivated by this, we develop the general theory of multipliers a little further in the next section. \chapter{Conclusion}\label{ccl} In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and should not be applied in the cases that are the exceptions to the rules. \\ \noindent The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy. The latter applies normally to a form of argument that is not a genuine rule of logic, where the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (such as the introduction of Pasch's axiom of Euclidean geometry and the five color theorem of graph theory). Pseudaria, an ancient lost book of false proofs, is attributed to Euclid. \\ \noindent Mathematical fallacies exist in many branches of mathematics. 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