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% need for subequations\\n\\\\usepackage{graphicx} % need for \
figures\\n\\\\usepackage{verbatim} % useful for program \
listings\\n\\\\usepackage{color} % use if color is used in \
text\\n\\\\usepackage{subfigure} % use for side-by-side \
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spacing between lines\\n\\n\\\\begin{comment}\\n\\\\pagestyle{empty} % use if \
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2006}\\n\\n\\\\maketitle\\n\\n\\\\section{Introduction}\\n\\\\TeX\\\\ looks \
more difficult than it is. It is\\nalmost as easy as $\\\\pi$. See how easy \
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$\\\\ldots$ We also can make\\nsubscripts\\n$A_{x}$, $A_{xy}$ and \
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spaces make no\\ndifference. Note that all formulas are enclosed by\\n\\\\$ \
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\\\\texttt{monospaced} fonts.\\n\\n\\\\section{Equations}\\nLet us see how \
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=\\\\sum_{i=1}^N w_i (x_i - \\\\bar{x})^2 .\\n\\\\end{equation}\\nIt is a \
good idea to number equations, but we can have a\\nequation without a number \
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s_i s_{i+1} ,\\n\\\\end{equation}\\nEquation~\\\\eqref{eq:ising} expresses \
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is necessary to process (typeset) a\\nfile twice to get the counters \
correct.}\\n\\nWe can define our own macros to save typing. For example, \
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\\\\newcommand{\\\\rb}{{\\\\rangle}}\\n\\\\end{verbatim}\\n\\\\newcommand{\\\\\
lb}{{\\\\langle}}\\n\\\\newcommand{\\\\rb}{{\\\\rangle}}\\nThen we can write \
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3\\n\\\\end{equation}\\n\\\\end{verbatim}\\nThe result \
is\\n\\\\begin{equation}\\n\\\\lb x \\\\rb = 3 \
.\\n\\\\end{equation}\\n\\nExamples of more complicated \
equations:\\n\\\\begin{equation}\\nI = \\\\! \\\\int_{-\\\\infty}^\\\\infty \
f(x)\\\\,dx \\\\label{eq:fine}.\\n\\\\end{equation}\\nWe can do some fine \
tuning by adding small amounts of horizontal\\nspacing:\\n\\\\begin{verbatim}\
\\n \\\\, small space \\\\! negative space\\n\\\\end{verbatim}\\nas is \
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different cases:\\n\\\\begin{equation}\\n\\\\label{eq:mdiv}\\nm(T) \
=\\n\\\\begin{cases}\\n0 & \\\\text{$T > T_c$} \\\\\\\\\\n\\\\bigl(1 - \
[\\\\sinh 2 \\\\beta J]^{-4} \\\\bigr)^{\\\\! 1/8} & \\\\text{$T < T_c$}\\n\\\
\\end{cases}\\n\\\\end{equation}\\nwrite \
matrices\\n\\\\begin{align}\\n\\\\textbf{T} &=\\n\\\\begin{pmatrix}\\nT_{++} \
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e^{\\\\beta (J - B)} \\\\hfill\\n\\\\end{pmatrix}.\\n\\\\end{align}\\nand \\n\
\\\\newcommand{\\\\rv}{\\\\textbf{r}}\\n\\\\begin{equation}\\n\\\\sum_i \
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\\\\mathbf{n}}\\\\, dA = P\\\\!\\\\int \\\\! {\\\\vec \\\\nabla} \\\\cdot \
\\\\rv\\\\, dV.\\n\\\\end{equation}\\n\\n\\\\section{Tables}\\nTables are a \
little more difficult. TeX\\nautomatically calculates the width of the \
columns.\\n\\n\\\\begin{table}[h]\\n\\\\begin{center}\\n\\\\begin{tabular}{|l|\
l|r|l|}\\n\\\\hline\\nlattice & $d$ & $q$ & $T_{\\\\rm mf}/T_c$ \\\\\\\\\\n\\\
\\hline\\nsquare & 2 & 4 & 1.763 \\\\\\\\\\n\\\\hline\\ntriangular & 2 & 6 & \
1.648 \\\\\\\\\\n\\\\hline\\ndiamond & 3 & 4 & 1.479 \
\\\\\\\\\\n\\\\hline\\nsimple cubic & 3 & 6 & 1.330 \
\\\\\\\\\\n\\\\hline\\nbcc & 3 & 8 & 1.260 \\\\\\\\\\n\\\\hline\\nfcc & 3 & \
12 & 1.225 \
\\\\\\\\\\n\\\\hline\\n\\\\end{tabular}\\n\\\\caption{\\\\label{tab:5/tc}\
Comparison of the mean-field predictions\\nfor the critical temperature of \
the Ising model with exact results\\nand the best known estimates for \
different spatial dimensions $d$\\nand lattice symmetries.}\\n\\\\end{center}\
\\n\\\\end{table}\\n\\n\\\\section{Lists}\\n\\nSome example of formatted \
lists include the\\nfollowing:\\n\\n\\\\begin{enumerate}\\n\\n\\\\item \
bread\\n\\n\\\\item cheese\\n\\n\\\\end{enumerate}\\n\\n\\\\begin{itemize}\\n\
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Dick\\n\\n\\\\end{itemize}\\n\\n\\\\section{Figures}\\n\\nWe can make figures \
bigger or smaller by scaling them. Figure~\\\\ref{fig:lj}\\nhas been scaled \
by 60\\\\%.\\n\\n\\\\begin{figure}[h]\\n\\\\begin{center}\\n\\\\\
includegraphics{figures/sine}\\n\\\\caption{\\\\label{fig:typical}Show me a \
sine.}\\n\\\\end{center}\\n\\\\end{figure}\\n\\n\\\\begin{figure}[h]\\n\\\\\
begin{center}\\n\\\\scalebox{0.6}{\\\\includegraphics{figures/lj}}\\n\\\\\
caption{\\\\label{fig:lj}Plot of the\\nLennard-Jones potential\\n$u(r)$. The \
potential is characterized by a length\\n$\\\\sigma$ and an \
energy\\n$\\\\epsilon$.}\\n\\\\end{center}\\n\\\\end{figure}\\n\\n\\\\section{\
Literal text}\\nIt is desirable to print program code exactly as it is typed \
in a\\nmonospaced font. Use \\\\verb \\\\begin{verbatim} and\\n\\\\verb \
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y0 = 10; // example of declaration and assignment statement\\ndouble v0 = 0; \
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step\\ndouble y = y0;\\n\\\\end{verbatim}\\nThe command \\\\verb \
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Symbols}\\n\\n\\\\subsection{Common Greek letters}\\n\\nThese commands may be \
used only in math mode. Only the most common\\nletters are included \
here.\\n\\n$\\\\alpha, \\n\\\\beta, \\\\gamma, \
\\\\Gamma,\\n\\\\delta,\\\\Delta,\\n\\\\epsilon, \\\\zeta, \\\\eta, \
\\\\theta, \\\\Theta, \\\\kappa,\\n\\\\lambda, \\\\Lambda, \\\\mu, \\\\nu,\\n\
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\\\\Omega$\\n\\n\\\\subsection{Special symbols}\\n\\nThe derivative is \
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\\\\to\\nx_{0}\\n\\\\end{equation}\\n\\\\begin{equation}\\nf(x) \\\\mathop \
{\\\\longrightarrow}\\n\\\\limits_{x \\\\to x_0} \
y\\n\\\\end{equation}\\n\\n\\\\noindent Order of \
magnitude:\\n\\\\begin{equation}\\n\\\\log_{10}f \\\\simeq \
n\\n\\\\end{equation}\\n\\\\begin{equation}\\nf(x)\\\\sim \
10^{n}\\n\\\\end{equation}\\nApproximate \
equality:\\n\\\\begin{equation}\\nf(x)\\\\simeq g(x)\\n\\\\end{equation}\\n\\\
\\LaTeX\\\\ is simple if we keep everything in \
proportion:\\n\\\\begin{equation}\\nf(x) \\\\propto x^3 .\\n\\\\end{equation}\
\\n\\nFinally we can skip some space by using commands such \
as\\n\\\\begin{verbatim}\\n\\\\bigskip \\\\medskip \\\\smallskip \
\\\\vspace{1pc}\\n\\\\end{verbatim}\\nThe space can be \
negative.\\n\\n\\\\section{Use of Color}\\n\\n{\\\\color{blue}{We can change \
colors for emphasis}},\\n{\\\\color{green}{but}} {\\\\color{cyan}{who is \
going pay for the \
ink?}}\\n\\n\\\\section{\\\\label{morefig}Subfigures}\\n\\nAs soon as many \
students start becoming comfortable using \\\\LaTeX, they want\\nto use some \
of its advanced features. So we now show how to place two\\nfigures side by \
side.\\n\\n\\\\begin{figure}[h!]\\n\\\\begin{center}\\n\\\\subfigure[Real and \
imaginary.]{\\n\\\\includegraphics[scale=0.5]{figures/reim}}\\n\\\\subfigure[\
Amplitude and \
phase.]{\\n\\\\includegraphics[scale=0.5]{figures/phase}}\\n\\\\caption{\\\\\
label{fig:qm/complexfunctions} Two representations of complex\\nwave \
functions.}\\n\\\\end{center}\\n\\\\end{figure}\\n\\nWe first have to include \
the necessary package,\\n\\\\verb+\\\\usepackage{subfigure}+, which has to go \
in the preamble (before\\n\\\\verb+\\\\begin{document}+). It sometimes can be \
difficult to place a figure in\\nthe desired place.\\n\\nYour LaTeX document \
can be easily modified to make a poster or a screen\\npresentation similar to \
(and better than) PowerPoint. Conversion to HTML is\\nstraightforward. \
Comments on this tutorial are appreciated.\\n\\n\\\\begin{thebibliography}{5}\
\\n\\n\\\\bibitem{latex}Helmut Kopka and Patrick W. Daly, \\\\textsl{A Guide \
to\\n\\\\LaTeX: Document Preparation for Beginners and Advanced \
Users},\\nfourth edition, Addison-Wesley \
(2004).\\n\\n\\\\bibitem{website}Some useful links are\\ngiven at \
\\\\url{}.\\n\\n\\\\end{thebibliography}\
\\n\\n{\\\\small \\\\smallskip\\\\noindent Updated 5 December \
2006.}\\n\\\\end{document}\>\""}], ";"}]], "Input",
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"\"\<\\\\documentclass[11pt,twoside]{book}\\n\\\\usepackage{etex}\\n\\\\\
reserveinserts{28}\\n\\\\usepackage{latexsym}\\n\\\\usepackage{amssymb}\\n\\\\\
usepackage{amsmath}\\n\\\\usepackage{amsthm}\\n\\\\usepackage{mathrsfs}\\n\\\\\
usepackage{graphicx}\\n\\\\usepackage{makeidx}\\n\\\\usepackage{fancyhdr}\\n\\\
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\\n\\\\makeindex\\n\\n\\n%\\\\doublespacing\\n\\\\begin{document}\\n\\n\\n\\\\\
frontmatter\\n\\\\pagenumbering{roman} \\n\\\\title{Wavelets: \\\\\\\\\\n A \
Concise Course}\\n\\\\author{Amir-Homayoon Najmi \
\\\\footnote{\\\\textit{Amir-Homayoon Najmi completed the Mathematical Tripos \
at Cambridge university in 1977, was a Fulbright scholar at the Relativity \
Center of University of Texas at Austin in 1978. He obtained his D.Phil. at \
the Astrophysics Department of Oxford University in 1982 with a thesis on \
quantum field theory in curved space times. He taught at the physics and \
mathematics departments of the University of Utah 1982-86 and was a research \
physicist at the Shell Oil Bellaire Geophysical Research Center 1986-1990. \
Since 1990 he has been at Applied Physics Laboratory of the Johns Hopkins \
University and is currently in the Electromagnetic and Acoustics Group. He \
teaches in the Physics and the Electrical Engineering programs of the Johns \
Hopkins University Whiting School Master's degree for professionals.}}, M.A. \
(Cantab.), D.Phil. \
(Oxon.)}\\n\\n\\n\\\\maketitle\\n\\n\\\\normalsize\\n\\n\\n\\\\include{book-\
dedication}\\n\\\\include{book-preface}\\n\\\\include{book-acknowledgement}\\\
n\\n\\n\\\\thispagestyle{empty}\\n\\\\newpage\\n\\\\pagestyle{plain} \
\\n\\\\tableofcontents\\n\\\\listoftables\\n\\\\listoffigures\\n\\\\newpage\\\
n\\\\mainmatter\\n\\\\pagenumbering{arabic} \\n\\n\\\\theoremstyle{plain} \
\\\\newtheorem{thm}{Theorem}\\n\\\\theoremstyle{definition} \
\\\\newtheorem{defn}{Definition}\\n%\\\\theoremstyle{remark} \
\\\\newtheorem{rem}{thm}{Remark}\\n\\n\\n\\n\\n\\n\\n\\n%%%%%%%%%%%%%%%%%%%%%%\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\\n%\\n% \
CHAPTER \
1\\n%\\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\
%%%%%%%%%%%%%%%%%%%%\\n\\n\\n\\n\\n\\\\chapter{Analysis in Vector and \
Function Spaces}\\n\\\\label{Chapter_AlgebraAnalysis}\\n\\n\\n\\\\section{\
Introduction}\\n\\nIndependent variables of interest are either continuous or \
discrete. If we denote by $t$ a continuous variable, $-\\\\infty \
M$ which is a \
basis.\\n\\\\end{thm}\\n\\\\newtheorem{Dimension}{Theorem}\\n\\\\begin{thm}\\\
n\\\\label{Dimension}\\nThe number of elements in any basis of a finite \
dimensional vector space is the same as in any other basis. This number $N$ \
is the dimension\\\\index{dimension} of the corresponding vector space. In \
addition, every set of $N+1$ vectors in an $N$-dimensional vector space is \
linearly dependent. \
\\n\\\\end{thm}\\n\\n\\n\\n\\\\newtheorem{SpanOfVectors}{Definition}\\n\\\\\
begin{defn}\\nGiven a set of vectors $\\\\left\\\\{ {\\\\mathbf{x}_j \
:\\\\;j=1,\\\\ldots ,N} \\\\right\\\\}$ which are not necessarily linearly \
independent, and a field $\\\\mathbb{F}$, the Span of this set is defined by \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-2}\\nSpan\\\\left\\\\{ \
{\\\\mathbf{x}_j :\\\\;j=1,\\\\ldots ,N} \\n\\\\right\\\\}=\\\\left\\\\{ \
{\\\\sum\\\\limits_{j=1}^N {c_j \\\\mathbf{x}_j ,\\\\;\\\\forall c_j \\\\in \
\\\\mathbb{F}} } \\\\right\\\\} ~.\\n\\\\end{equation}\\n\\\\end{defn}\\nNote \
that the Span of a set of vectors satisfies all the conditions required for a \
vector space. So if the span of a set of vectors is not the entire space, \
then it is a proper subspace. \\n\\n\\\\newtheorem{InfiniteDimensional \
VectorSpace}{Definition}\\n\\\\begin{defn}\\nAn infinite dimensional vector \
space is a vector space in which no linear span of the form \
\\\\eqref{Chapter1-2} is the entire space, for any integer \
$N$.\\n\\\\end{defn}\\n\\\\noindent In other words, for any given integer \
$N$, however large, and any set of vectors $\\\\mathbf{x}_j, ~ j=1, \
\\\\ldots, N$, there is at least one vector $\\\\mathbf{y}$ that is not in \
the linear span of those $N$ vectors.\\n\\nWe continue to denote a vector in \
a finite dimensional vector space by bold letters, e.g., ${\\\\mathbf{x}}$. \
An infinite dimensional vector space in this text is a space of functions of \
a real variable, e.g. $x \\\\left( t \\\\right)$, which we often abbreviate \
to $x$, dropping the dependence on the real variable $t$. Although many \
definitions and properties in this chapter use bold letters that refer to \
vectors in finite dimensional vector spaces, most are equally applicable to \
infinite dimensional space of functions. The two notations are for the most \
part interchangeable unless otherwise stated. \
\\n\\n\\\\newtheorem{InnerAndDirectSum}{Definition}\\n\\\\begin{defn}\\n\\\\\
label{InnerAndDirectSum}\\nIf $\\\\mathbb{V}$ and $\\\\mathbb{W}$ are \
subspaces of the vector space $\\\\mathscr{V}$, the inner sum, denoted by \
$\\\\mathbb{V+W}$, is a subspace defined by \\n\\\\[\\n\\\\mathbb{V+W} \
\\\\equiv \\\\left\\\\{ {\\\\mathbf{u}\\\\in \
\\\\mathscr{V}:\\\\;\\\\mathbf{u}=\\\\mathbf{v+w};\\\\;\\\\mathbf{v}\\\\in \\\
\\mathbb{V},\\\\mathbf{w}\\\\in \\\\mathbb{W}} \\\\right\\\\}, \
\\n\\\\]\\nwhile the direct sum of the two subspaces, denoted by \
$\\\\mathbb{V}\\\\oplus \\\\mathbb{W}$, is defined by \
\\n\\\\[\\n\\\\mathbb{V}\\\\oplus \\\\mathbb{W} \\\\equiv \\\\left\\\\{ \
{\\\\left( {\\\\mathbf{v},\\\\mathbf{w}} \\\\right): ~ \\\\mathbf{v}\\\\in \\\
\\mathbb{V},\\\\mathbf{w}\\\\in \\\\mathbb{W}} \\\\right\\\\}. \
\\n\\\\]\\n\\\\end{defn}\\n\\n\\\\noindent If $\\\\mathbb{V}$ and \
$\\\\mathbb{W}$ are disjoint, i.e. $\\\\mathbb{V}\\\\cap \
\\\\mathbb{W}=\\\\left\\\\{ {\\\\mathbf{0}} \\\\right\\\\}$, then \
$\\\\mathbb{V+W}$ and $\\\\mathbb{V}\\\\oplus \\\\mathbb{W}$ are isomorphic, \
i.e. they have identical algebraic structure. \\n\\nIn order to do analysis \
we must be able to tell how close vectors are to each other, and so we \
introduce the concept of normed vector spaces in the next section \
\\\\cite{BaNa66}.\\n\\n\\\\section{Normed Vector \
Spaces}\\n\\n\\\\newtheorem{NormedSpace}{Definition}\\n\\\\begin{defn}\\nA \
Normed Vector Space is equipped with a non negative real valued function (the \
Norm), $\\\\left\\\\| {\\\\;\\\\cdot \\\\,} \
\\\\right\\\\|:\\\\mathscr{V}\\\\to {\\\\rm \\\\mathbb{R}}$, such \
that\\n\\\\begin{subequations}\\n\\\\label{Chapter1-3}\\n\\\\begin{align}\\n \
\\\\left\\\\| {\\\\mathbf{x}} \\\\right\\\\| & \\\\ge 0 ~ \\\\forall \
{x}\\\\in \\\\mathscr{V} ~, \\\\label{Chapter1-3a}\\\\\\\\ \\n \\\\left\\\\| \
{\\\\mathbf{x}} \\\\right\\\\| & = 0\\\\Leftrightarrow \\\\mathbf{x=0} ~, \
\\\\label{Chapter1-3b}\\\\\\\\ \\n \\\\left\\\\| {a \\\\mathbf{x}} \
\\\\right\\\\| & =\\\\left| a \\\\right|\\\\left\\\\| {\\\\mathbf{x}} \
\\\\right\\\\| ~, \\\\label{Chapter1-3c}\\\\\\\\ \\n \\\\left\\\\| \
\\\\mathbf{x+y} \\\\right\\\\| & \\\\le \\\\left\\\\| {\\\\mathbf{x}} \
\\\\right\\\\|+\\\\left\\\\| {\\\\mathbf{y}} \\n\\\\right\\\\|\\\\;\\\\quad \
\\\\forall \\\\mathbf{x},\\\\mathbf{y}\\\\in \\\\mathscr{V} \
\\\\label{Chapter1-3d} ~. \\n \
\\\\end{align}\\n\\\\end{subequations}\\n\\\\end{defn}\\n\\\\noindent \
Condition \\\\eqref{Chapter1-3d} is known as the triangle \
inequality\\\\index{triangle inequality}. For instance, if $\\\\mathscr{V}$ \
is the infinite dimensional vector space of all square-integrable functions \
defined on the interval $\\\\left[ {a,b} \\\\right]\\\\in {\\\\rm \
\\\\mathbb{R}}$, the $L_2$ norm of a function ${x\\\\left( t \\\\right)}$ is \
defined by \\n\\\\[\\n\\\\left\\\\| {x\\\\left( t \\\\right)} \
\\\\right\\\\|_2 =\\\\left[ {\\\\int_a^b {\\\\left| {x\\\\left( t \\\\right)} \
\\\\right|^2dt} } \\\\right]^{{1 \\\\mathord{\\\\left/\\n {\\\\vphantom {1 \
2}} \\\\right. \\\\kern-\\\\nulldelimiterspace} 2}} ~. \\n\\\\]\\nThis can be \
generalized to an $L_p $ norm where $1\\\\le p<\\\\infty $ and \
$\\\\mathscr{V}$ is the vector space of all functions whose magnitude raised \
to the power $p$ is integrable\\n\\\\[\\n\\\\left\\\\| {x\\\\left( t \
\\\\right)} \\\\right\\\\|_p =\\\\left[ {\\\\int_a^b {\\\\left| {x\\\\left( t \
\\\\right)} \\\\right|^pdt} } \\\\right]^{{1 \\\\mathord{\\\\left/\\n \
{\\\\vphantom {1 p}} \\\\right. \\\\kern-\\\\nulldelimiterspace} p}} \
~.\\n\\\\]\\n\\nThe norm enables us to define convergence in a normed vector \
space. In the infinite dimensional space $L_{2}(\\\\mathbb{R})$, the space \
of functions with finite $L_2$ norm, three types of convergence can be \
defined. \
\\n\\\\newtheorem{ConvergenceInTheMean}{Definition}\\n\\\\begin{defn}\\n\\\\\
label{ConvergenceInTheMean}\\nA sequence $x_{n}(t)$, $n = 0, 1, 2, \\\\ldots$ \
converges in the mean to the function $x(t)$ if $\\\\left\\\\| {x_n - x} \
\\\\right\\\\| \\\\to 0$ for \\n$n \\\\to \\\\infty$. \
\\n\\\\end{defn}\\nUnless otherwise stated, this is the definition for \
convergence that is used throughout this book. Here we will state the other \
two possible definitions for \
convergence.\\n\\\\newtheorem{ConvergencePointwise}{Definition}\\n\\\\begin{\
defn}\\nA sequence $x_{n}(t)$, $n = 0, 1, 2, \\\\ldots$ converges pointwise \
to the function $x(t)$ if given the point $t$ and a small positive number $\\\
\\varepsilon$, an integer $N_{t}$ can be found for which $\\\\left| {x_n \
\\\\left( t \\\\right) - x\\\\left( t \\\\right)} \\\\right| < \\\\varepsilon \
$\\nfor all $n \\\\ge \
N_t$.\\n\\\\end{defn}\\n\\\\newtheorem{ConvergenceUniform}{Definition}\\n\\\\\
begin{defn}\\nA sequence $x_{n}(t)$, $n = 0, 1, 2, \\\\ldots$ converges \
uniformly to the function $x(t)$ if for any small positive number \
$\\\\varepsilon$, an integer $N$ can be found that is independent of $t$ and \
for which $\\\\left| {x_n \\\\left( t \\\\right) - x\\\\left( t \\\\right)} \
\\\\right| < \\\\varepsilon $\\nfor all $n \\\\ge N$.\\n\\\\end{defn}\\nGiven \
a sequence of functions $x_{n}(t)$ converging in the mean to the function \
$x(t)$, and using the triangle inequality we have\\n\\\\[\\n\\\\left\\\\| \
{x_n - x_m } \\\\right\\\\| = \\\\left\\\\| {x_n - x + x - x_m } \
\\\\right\\\\| \\\\le \\\\left\\\\| {x_n - x} \\\\right\\\\| + \\\\left\\\\| \
{x_m - x} \\\\right\\\\|~,\\n\\\\]\\nwhich in view of the convergence \
property, can be made as small as we please by taking the integers $n$ and \
$m$ sufficiently large. This motivates the following \
definition.\\n\\\\newtheorem{CauchySequence}{Definition}\\n\\\\begin{defn}\\n\
\\\\label{CauchySequence}\\nA sequence $x_{n}(t)$, $n = 0, 1, 2, \\\\ldots $ \
is called a Cauchy sequence\\\\index{Cauchy sequence} if, for any \
$\\\\varepsilon > 0$ there is a positive integer $N$ (which could depend on $\
\\\\varepsilon$) such that $\\\\left\\\\| {x_n - x_m } \\\\right\\\\| < \
\\\\varepsilon $ for $n,m \\\\ge N$.\\n\\\\end{defn}\\n\\\\noindent All \
convergent sequences are Cauchy sequences, but the converse is not true.\\n\\\
\\newtheorem{ClosedSubspace}{Definition}\\n\\\\begin{defn}\\n\\\\label{\
ClosedSubspace}\\nA subspace of a normed vector space is closed if all \
convergent sequences in that subspace converge to points within that \
subspace.\\n\\\\end{defn}\\n\\n\\n\\n\\\\section{Inner Product}\\nAn inner \
product\\\\index{inner product} can be used to project vectors in arbitrary \
directions, in addition to introducing the concept of orthogonality between \
vectors. \\n\\\\newtheorem{InnerProduct}{Definition}\\n\\\\begin{defn}\\nThe \
inner product is a complex function $<\\\\cdot ,\\\\cdot >: \
\\\\mathscr{V}\\\\times \\\\mathscr{V}\\\\to {\\\\rm \\\\mathbb{C}}$ with the \
following properties \\\\footnote{The space $\\\\mathscr{V} \\\\times \
\\\\mathscr{V}$ denotes the set of all ordered pairs $\\\\left( \
{\\\\mathbf{x},\\\\mathbf{y}} \\\\right)$ where $\\\\mathbf{x},\\\\mathbf{y} \
\\\\in \\\\mathscr{V}$.} \
\\n\\\\begin{subequations}\\n\\\\label{Chapter1-4}\\n\\\\begin{align}\\n \
\\\\left\\\\langle {\\\\mathbf{x},\\\\mathbf{y}} \\\\right\\\\rangle & = \
\\\\left\\\\langle {\\\\mathbf{y},\\\\mathbf{x}} \\\\right\\\\rangle ^\\\\ast \
\\\\label{Chapter1-4a} \\\\\\\\ \\n \\\\left\\\\langle \
{\\\\mathbf{x}},a\\\\,{\\\\mathbf{y}} \\\\right\\\\rangle & = a \
\\\\left\\\\langle {\\\\mathbf{x},\\\\mathbf{y}} \\\\right\\\\rangle \
\\\\label{Chapter1-4b} \\\\\\\\ \\n \\\\left\\\\langle \
{\\\\mathbf{x+y},\\\\mathbf{z}} \\\\right\\\\rangle & = \\\\left\\\\langle \
{\\\\mathbf{x},\\\\mathbf{z}} \\\\right\\\\rangle + \\\\left\\\\langle \
{\\\\mathbf{y},\\\\mathbf{z}} \\\\right\\\\rangle \
\\\\label{Chapter1-4c}\\\\\\\\ \\n \\\\left\\\\langle\\\\ \
{\\\\mathbf{x},\\\\mathbf{x}} \\\\right\\\\rangle \\\\;\\\\; & > 0,\\\\quad \
\\\\forall \\\\mathbf{x}\\\\ne \\\\mathbf{0} \\\\label{Chapter1-4d} \\\\\\\\ \
\\n \\\\left\\\\langle {\\\\mathbf{x},\\\\mathbf{x}} \\\\right\\\\rangle & = \
0\\\\Leftrightarrow \\\\mathbf{x}=\\\\mathbf{0} \
\\\\label{Chapter1-4e}\\n\\\\end{align}\\n\\\\end{subequations}\\n\\\\end{\
defn}\\nFor example, if $\\\\mathscr{V}={\\\\rm \\\\mathbb{R}}^n$ and \
$\\\\mathbf{x}=\\\\left[ {{\\\\begin{array}{*{20}c} {x_1 } \\\\hfill & {x_2 } \
\\\\hfill & \\\\cdots \\\\hfill & {x_n } \\\\hfill \\\\\\\\\\\\end{array} }} \
\\\\right]^T,\\\\,x_j \\\\in {\\\\rm \\\\mathbb{R}}$ and superscript $T$ \
denotes transposition, then \\n\\\\begin{equation}\\n\\\\label{Chapter1-5}\\n\
\\\\left\\\\langle {\\\\mathbf{x},\\\\mathbf{y}} \\\\right\\\\rangle = \
\\\\mathbf{x}\\\\cdot \\\\mathbf{y}=\\\\mathbf{x}^T \
\\\\mathbf{y}=\\\\sum\\\\limits_{j=1}^n {x_j y_j } \\n\\\\end{equation}\\nIf, \
on the other hand, $\\\\mathscr{V}={\\\\rm \\\\mathbb{C}}^n$ and $x_j \\\\in \
{\\\\rm \\\\mathbb{C}}$, \
then\\n\\\\begin{equation}\\n\\\\label{Chapter1-6}\\n\\\\left\\\\langle \
{\\\\mathbf{x},\\\\mathbf{y}} \\\\right\\\\rangle=\\\\mathbf{x}^\\\\ast \
\\\\cdot \\\\mathbf{y}=\\\\mathbf{x}^H \
\\\\mathbf{y}=\\\\sum\\\\limits_{j=1}^n {x_{_j }^\\\\ast y_j} \
\\n\\\\end{equation}\\nThe superscript $H$ denotes the Hermitian conjugate, $\
\\\\mathbf{x}^H\\\\equiv \\\\left[{{\\\\begin{array}{*{20}c}\\n {x_1^\\\\ast \
} \\\\hfill & \\\\cdots \\\\hfill & {x_n^\\\\ast } \\\\hfill \
\\\\\\\\\\\\end{array} }} \\\\right]$. Note that the relation \
\\n\\\\[\\n\\\\left\\\\langle {a \\\\mathbf{x},\\\\mathbf{y}} \
\\\\right\\\\rangle =a^\\\\ast \\\\left\\\\langle \
{\\\\mathbf{x},\\\\mathbf{y}} \\\\right\\\\rangle\\n\\\\]\\nis a consequence \
of the properties \\\\eqref{Chapter1-4a} and \\\\eqref{Chapter1-4b} of the \
inner product. The usual example of an inner product on a function space is \
\\n\\\\[\\n\\\\left\\\\langle {x,y} \\\\right\\\\rangle = \
\\\\int\\\\limits_{ - \\\\infty }^\\\\infty {x^ * \\\\left( t \\\\right)} y\
\\\\left( t \\\\right)dt\\n\\\\]\\nwhere and $x$ and $y$ are square \
integrable complex valued functions of the continuous variable $t \\\\in \
\\\\mathbb{R}$.\\n\\nAn inner product space has a natural norm, called the \
induced norm\\\\index{norm}, associated with the inner \
product\\n\\\\[\\n\\\\left\\\\| {\\\\mathbf{x}} \\\\right\\\\|\\\\equiv \
\\\\sqrt {\\\\left\\\\langle {\\\\mathbf{x},\\\\mathbf{x}} \
\\\\right\\\\rangle} ~.\\n\\\\]\\nThere is a converse to the above relation \
in the sense that the knowledge of the norm in an inner product space is \
sufficient to recover the associated inner product, as shown in the following \
theorem known as the Polarization \
identity.\\n\\\\newtheorem{PolarizationId}{Theorem}\\n\\\\begin{thm}\\n\\\\\
label{PolarizationId}\\nLet $\\\\mathscr{V}$ be an inner product space and \
let $\\\\mathbf{x}$ and $\\\\mathbf{y} \\\\in \\\\mathscr{V}$. Then, if \
$\\\\mathscr{V}$ is a real space\\n\\\\[\\n\\\\left\\\\langle {\\\\mathbf{x},\
\\\\mathbf{y}} \\\\right\\\\rangle \\\\equiv \\\\frac{1}{4}\\\\left\\\\| {\\\
\\mathbf{x} + \\\\mathbf{y}} \\\\right\\\\|^2 - \\\\frac{1}{4}\\\\left\\\\| \
{\\\\mathbf{x} - \\\\mathbf{y}} \\\\right\\\\|^2 ~,\\n\\\\]\\nand if \
$\\\\mathscr{V}$ is a complex space\\n\\\\[\\n\\\\left\\\\langle \
{\\\\mathbf{x},\\\\mathbf{y}} \\\\right\\\\rangle \\\\equiv \\\\frac{1}{4}\\\
\\left\\\\| {\\\\mathbf{x} + \\\\mathbf{y}} \\\\right\\\\|^2 - \
\\\\frac{1}{4}\\\\left\\\\| {\\\\mathbf{x} - \\\\mathbf{y}} \\\\right\\\\|^2 \
+ \\\\frac{i}{4}\\\\left\\\\| {\\\\mathbf{x} + i\\\\mathbf{y}} \
\\\\right\\\\|^2 - \\\\frac{i}{4}\\\\left\\\\| {\\\\mathbf{x} - \
i\\\\mathbf{y}} \\\\right\\\\|^2 ~.\\n\\\\]\\n\\\\end{thm}\\nThe induced norm \
satisfies the Cauchy-Schwartz\\\\index{Cauchy-Schwartz} inequality stated in \
the following theorem. \
\\n\\\\newtheorem{CauchySchwartz}{Theorem}\\n\\\\begin{thm}\\n\\\\label{\
CauchySchwartz}\\nGiven an inner product space $\\\\mathscr{V}$ and two \
elements $\\\\mathbf{x}$ and $\\\\mathbf{y}$, then\\n\\\\begin{equation}\\n\\\
\\label{Chapter1-7}\\n\\\\left| {\\\\left\\\\langle \
{\\\\mathbf{x},\\\\mathbf{y}} \\\\right\\\\rangle} \\\\right|\\\\le \
\\\\left\\\\| {\\\\mathbf{x}} \\n\\\\right\\\\|\\\\left\\\\| {\\\\mathbf{y}} \
\\\\right\\\\|\\n\\\\end{equation}\\nwith equality if, and only if, \
$\\\\mathbf{y}=a \\\\mathbf{x}$ for some (complex) constant \
$a$.\\n\\\\end{thm}\\nTwo vectors are orthogonal if their inner product is \
zero: $\\\\mathbf{x}\\\\bot \\\\mathbf{y}\\\\Leftrightarrow \
\\\\left\\\\langle {\\\\mathbf{x},\\\\mathbf{y}} \\\\right\\\\rangle=0$. One \
can, using the Cauchy-Schwartz inequality, define an angle $\\\\theta$ \
between two vectors by \\n\\\\begin{equation}\\n\\\\label{Chapter1-8}\\ncos\\\
\\left( \\\\theta \\\\right)\\\\equiv \\\\frac{\\\\left| {\\\\left\\\\langle \
{\\\\mathbf{x},\\\\mathbf{y}} \\\\right\\\\rangle} \\\\right|}{\\\\left\\\\| \
{\\\\mathbf{x}} \\\\right\\\\|\\\\left\\\\| {\\\\mathbf{y}} \
\\n\\\\right\\\\|}, ~~ 0 \\\\le \\\\theta \\\\le \\\\frac{\\\\pi }{2} ~.\\n\\\
\\end{equation}\\n\\nThe concept of orthogonality can be extended to \
subspaces of an inner product vector space. \
\\n\\\\newtheorem{Vperp}{Definition}\\n\\\\begin{defn}\\n\\\\label{Vperp}\\\
nGiven a subspace $\\\\mathbb{V}$ of an inner product space $\\\\mathscr{V}$, \
the space $\\\\mathbb{V} ^\\\\bot$ is defined to contain all vectors that are \
orthogonal to every vector in $\\\\mathbb{V}$. That is, \
$<\\\\mathbf{v},\\\\mathbf{w}>=0$ whenever $\\\\mathbf{v}\\\\in \
\\\\mathbb{V}$ and $\\\\mathbf{w}\\\\in \\\\mathbb{V} ^ \
\\\\bot$.\\n\\\\end{defn}\\n\\n\\n\\\\newtheorem{VperpSubspace}{Theorem}\\n\\\
\\begin{thm}\\n\\\\label{VperpSubspace}\\n$\\\\mathbb{V} ^\\\\bot$ is a \
subspace of $\\\\mathscr{V}$ and $\\\\mathbb{V} \\\\subset \\\\mathbb{V}^{ \\\
\\bot \\\\bot }$ and $\\\\mathbb{V}^ \\\\bot = \\\\mathbb{V}^{ \\\\bot \\\
\\bot \\\\bot }$.\\n\\\\end{thm}\\n\\nIn transform theory a function $x(t)$ \
in $L_2(\\\\mathbb{R})$ is mapped to a set of transform coefficients by \
calculating the inner products of the given function with a set of basis \
function $\\\\xi_a(t)$. Using theorem \\\\ref{CauchySchwartz}, the \
magnitude squared of the inner product $\\\\left\\\\langle {\\\\xi _a ,x} \
\\\\right\\\\rangle$ satisfies the inequality \\n\\\\[\\n\\\\left| {\\\\left\
\\\\langle {\\\\xi_a ,x} \\\\right\\\\rangle } \\\\right|^2 \\\\le {\\\\left\
\\\\| {\\\\xi _a } \\\\right\\\\|}^2 {\\\\left\\\\| x \\\\right\\\\|}^2\\n\\\
\\]\\nand so the quantity $\\\\left| {\\\\left\\\\langle {\\\\xi_a ,x} \
\\\\right\\\\rangle } \\\\right|^2$ can be used to find the matching value \
$a$ of the basis function that $x(t)$ may be known to be proportional to. \
However, if the function of interest $x$ is not proportional to any of the \
basis functions, then the best match is found by minimizing the magnitude \
squared of the sum or the difference ${x \\\\pm \\\\xi_a }$. Thus, the \
minima of the \
quantities\\n\\\\begin{equation}\\n\\\\label{Chapter1-9}\\n\\\\left\\\\| {x \
\\\\pm \\\\xi_a } \\\\right\\\\|^2 = \\\\left\\\\| x \\\\right\\\\|^2 + \
\\\\left\\\\| {\\\\xi_a } \\\\right\\\\|^2 \\\\pm 2{\\\\mathop{\\\\rm \
Re}\\\\nolimits} \\\\left\\\\langle {\\\\xi _a ,x} \\\\right\\\\rangle \
~,\\n\\\\end{equation}\\noccur at the maximum of $\\\\left( {{\\\\rm \
Re}\\\\left\\\\langle {\\\\xi_a ,x} \\\\right\\\\rangle } \\\\right)^2$. \\n\
\\n\\n\\\\section{Banach and Hilbert Spaces}\\nAlthough in a normed space we \
have the means for doing analysis and study convergence, we are not \
guaranteed that converging sequences of vectors actually have a limit inside \
the space itself \\\\cite{BaNa66}. A Banach\\\\index{Banach} space is a \
normed space in which every convergent Cauchy sequence (see definition \
\\\\ref{CauchySequence}) of vectors converges to a vector inside the space. \
We take this as our working definition of completeness\\\\index{completeness} \
\\\\footnote{A more general definition in a metric space relies on first \
defining the diameter of closed subspaces and then requiring that every \
decreasing sequence of non empty closed subsets of the corresponding space \
whose diameters tend to $0$ must have a non-empty intersection, i.e., a point \
that is common to all of them.}.\\n\\\\newtheorem{BanachSpace}{Definition}\\n\
\\\\begin{defn}\\nA Banach space is a complete normed vector space. \
\\n\\\\end{defn}\\n\\\\newtheorem{HilbertSpace}{Definition}\\n\\\\begin{defn}\
\\nA Hilbert space $\\\\mathscr{H}$ is a complete inner product space, or \
equivalently, a Banach space whose norm is induced by an existing inner \
product, $\\\\left\\\\| x \\\\right\\\\| = \\\\sqrt {\\\\left\\\\langle {x,x} \
\\\\right\\\\rangle }$.\\n\\\\end{defn}\\nHilbert spaces commonly occur in \
mathematics and physics as function spaces. Their study is, therefore, \
contained in a branch of mathematics known as functional analysis. Hilbert \
spaces generalize the familiar Euclidean three dimensional space \
$\\\\mathbb{R}^3$: most important geometrical results in the latter have \
direct generalizations to Hilbert spaces. For instance, orthogonal \
projections from the tip of a standard three dimensional vector on to an \
arbitrary plane has its analogue in Hilbert spaces in the form of orthogonal \
projections onto linear subspaces and is at the foundation of linear optimal \
filter theory. Orthogonal projections are discussed in section \
\\\\ref{OrthogonalProjections}. \\n\\nAn indispensable property of Hilbert \
spaces in almost all of physics and applied mathematics is that of \
separability: separable Hilbert spaces\\\\index{Hilbert space!separable} \
possess a countable orthonormal basis (see section \
\\\\ref{CompleteOrthonormalBases} for orthonormal bases). The function space \
of interest to us is $L_2(\\\\mathbb{R})$ which is a separable Hilbert space. \
One such basis can be found starting with the windowed (or short time) \
Fourier transform and its discretization in both time and frequency. The \
resulting basis, although localized, have fixed time and frequency \
resolutions that are inversely proportional to each other. The continuous \
wavelet transform is another approach to arrive at variable resolution \
functions that are localized in time and scale. Both methods will be \
discussed in chapter \
\\\\ref{Chapter_TimeFrequencyScale}.\\n\\n\\n\\n\\n\\\\section{Linear \
Operators, Operator Norm, the Adjoint \
Operator}\\n\\n\\\\newtheorem{LinearOperator}{Definition}\\n\\\\begin{defn}\\\
nGiven two vector spaces $\\\\mathscr{V}_1$ and $\\\\mathscr{V}_2$, a \
transformation (operator) $T$ between the two denoted by $T: \\\\mathscr{V}_1 \
\\\\to \\\\mathscr{V}_2$ is said to be linear if for every $\\\\mathbf{x},\\\
\\mathbf{y} \\\\in \\\\mathscr{V}_1$ and all $\\\\alpha, \\\\beta \\\\in \
\\\\mathbb{C}$, $T\\\\left( {\\\\alpha \\\\mathbf{x} + \\\\beta \
\\\\mathbf{y}} \\\\right) = \\\\alpha T\\\\left( \\\\mathbf{x} \\\\right) + \
\\\\beta T\\\\left( \\\\mathbf{y} \\\\right)$. \\n\\\\end{defn}\\nFor \
instance, if the two vector spaces are $\\\\mathbb{C}^n$ and \
$\\\\mathbb{C}^m$, then $T$ is an $m \\\\times n$ matrix of complex numbers. \
When the underlying spaces are both copies of $L_2 \\\\left( \\\\mathbb{R} \\\
\\right)$, a linear transformation takes the form of an integral operator, \
e.g., $X\\\\left( \\\\omega \\\\right) = \\\\int\\\\limits_{ - \\\\infty }^\\\
\\infty {T\\\\left( {\\\\omega, t } \\\\right)} x\\\\left( t \\\\right)dt$. \
An example of such an operator is the Fourier transform (see equations \
\\\\eqref{Chapter1-56} and \\\\eqref{Chapter1-57}). \\n\\nIn order to ensure \
that the image of a linear operator\\\\index{operator!linear} is in fact a \
member of $L_2(\\\\mathbb{R})$ we must extend the notion of norm to linear \
operators. An operator norm\\\\index{operator!norm} must have all the \
required properties of a norm shown in equation \\\\eqref{Chapter1-3}. The \
$p$-norm of a linear operator $T:\\\\mathscr{V}_1 \\\\to \\\\mathscr{V}_2 $ \
is defined by \\\\footnote{The definition of equation \\\\eqref{Chapter1-10} \
is equivalent to $\\\\left\\\\| T \\\\right\\\\|_p = Max\\\\left\\\\{ \
{{{\\\\left\\\\| {Tx} \\\\right\\\\|_p } \\\\mathord{\\\\left/ {\\\\vphantom \
{{\\\\left\\\\| {Tx} \\\\right\\\\|_p } {\\\\left\\\\| x \\\\right\\\\|_p }}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} {\\\\left\\\\| x \\\\right\\\\|_p \
}};~x \\\\in \\\\mathscr{V}_1 ,\\\\left\\\\| x \\\\right\\\\|_p \\\\ne 0} \
\\\\right\\\\}$.}\\n\\\\begin{equation}\\n\\\\label{Chapter1-10}\\n\\\\left\\\
\\| T \\\\right\\\\|_p = Max \\\\left\\\\{ {\\\\left\\\\| {Tx} \
\\\\right\\\\|_p ; ~ x \\\\in \\\\mathscr{V}_1 {\\\\rm ~ and ~ }\\\\left\\\\| \
x \\\\right\\\\|_p = 1} \\\\right\\\\}\\n\\\\end{equation}\\nwhere $\\\\left\
\\\\| x \\\\right\\\\|_p$ is the $L_p$ norm defined previously. The $p$-norm \
of an operator satisfies the following inequalities \\n\\\\begin{equation}\\n\
\\\\label{Chapter1-11}\\n\\\\left\\\\| {Tx} \\\\right\\\\|_p \\\\le \\\\left\
\\\\| T \\\\right\\\\|_p \\\\left\\\\| x \\\\right\\\\|_p {\\\\rm and \
}\\\\left\\\\| {T_1 T_2 } \\\\right\\\\|_p \\\\le \\\\left\\\\| {T_1 } \
\\\\right\\\\|_p \\\\left\\\\| {T_2 } \\\\right\\\\|_p ~.\\n\\\\end{equation}\
\\nWhen the operator is defined $T:\\\\mathscr{V} \\\\to \\\\mathscr{V}$ and \
it has an inverse we can \
show\\n\\\\begin{equation}\\n\\\\label{Chapter1-12}\\n\\\\left\\\\| {T^{ - 1} \
} \\\\right\\\\|_p = \\\\frac{1}{{Min\\\\left\\\\{ {\\\\left\\\\| {Tx} \
\\\\right\\\\|_p ;\\\\left\\\\| x \\\\right\\\\|_p = 1,x \\\\in \
\\\\mathscr{V}} \
\\\\right\\\\}}}\\n\\\\end{equation}\\n\\\\newtheorem{BoundedOp}{Definition}\\\
n\\\\begin{defn}\\nA bounded operator\\\\index{operator!bounded} is a linear \
operator with finite norm. \
\\n\\\\end{defn}\\n\\\\newtheorem{BoundedOperator}{Theorem}\\n\\\\begin{thm}\\\
n\\\\label{BoundedOperator}\\nA linear operator is bounded if, and only if, \
it is continuous.\\n\\\\end{thm}\\nThe following theorem describes a useful \
formula, known as the Neumann expansion\\\\index{Neumann expansion}, for \
bounded operators.\\n\\\\newtheorem{NeumannExpansion}{Theorem}\\n\\\\begin{\
thm}\\n\\\\label{NeumannExpansion}\\nFor a bounded operator $T$ with unity \
bound, i.e., $\\\\left\\\\| T \\\\right\\\\| < 1$, where the norm satisfies \
the inequalities \\\\eqref{Chapter1-11}, we have the following formula \
\\\\cite{MaWa64}.\\n\\\\begin{equation}\\n\\\\label{Chapter1-13}\\n\\\\left( \
{1-T} \\\\right)^{-1} = \\\\sum\\\\limits_{k = 0}^\\\\infty {T^k} \
~.\\n\\\\end{equation}\\n\\\\end{thm}\\n\\nEvery bounded linear operator \
$T:\\\\mathscr{H}_1 \\\\to \\\\mathscr{H}_2 $ between two Hilbert spaces \
$\\\\mathscr{H}_1$ and $\\\\mathscr{H}_2$, has an associated \
adjoint\\\\index{operator!adjoint} $T^ + :\\\\mathscr{H}_2 \\\\to \
\\\\mathscr{H}_1 \
$.\\n\\\\newtheorem{AdjointdOperator}{Definition}\\n\\\\begin{defn}\\n\\\\\
label{AdjointOperator}\\nThe adjoint of a bounded linear operator \
$T:\\\\mathscr{H}_1 \\\\to \\\\mathscr{H}_2 $ is a linear operator $T^ + \
:\\\\mathscr{H}_2 \\\\to \\\\mathscr{H}_1 $ defined by the \
equation\\n\\\\begin{equation}\\n\\\\label{Chapter1-14}\\n\\\\left\\\\langle \
{s_2,T s_1} \\\\right\\\\rangle \\\\equiv \\\\left\\\\langle {T^ + s_2, \
s_1} \\\\right\\\\rangle ~,\\n\\\\end{equation}\\nfor every $s_1 \\\\in \
\\\\mathscr{H}_1$ and $s_2 \\\\in \\\\mathscr{H}_2$. When the two Hilbert \
spaces are the same, $\\\\mathscr{H}_1 = \\\\mathscr{H}_2 = \\\\mathscr{H}$, \
a linear bounded operator is self adjoint if $T = T^ +$. \
\\n\\\\end{defn}\\nTo every linear operator $T:\\\\mathscr{H}_1 \\\\to \
\\\\mathscr{H}_2 $ there correspond two fundamental linear subspaces, namely, \
the range space\\\\index{operator!range space} $\\\\mathscr{R}_T $ and the \
null space\\\\index{operator!null space} $\\\\mathscr{N}_T$. \
\\n\\\\newtheorem{RangeNullSpace}{Definition}\\n\\\\begin{defn}\\nThe range \
space of a linear operator $T:\\\\mathscr{H}_1 \\\\to \\\\mathscr{H}_2 $ is \
the image of the linear map and it is a linear subspace of \
$\\\\mathscr{H}_2$: \\n\\\\[\\n\\\\mathscr{R}_T =T\\\\left( {\\\\mathscr{H}_1 \
} \\\\right)=\\\\left\\\\{ {\\\\mathbf{y}\\\\,\\\\in \\\\mathscr{H}_2 \
:\\\\,\\\\mathbf{y}=T\\\\mathbf{x},\\\\mbox{ for some }\\\\mathbf{x}\\\\in \\\
\\mathscr{H}_1 \\\\mbox{ }} \\\\right\\\\}. \\n\\\\]\\nThe null space is a \
linear subspace of $\\\\mathscr{H}_1 $ and contains all elements of \
$\\\\mathscr{H}_1 $ that map to the zero of $\\\\mathscr{H}_2 $: \
\\n\\\\[\\n\\\\mathscr{N}_T=\\\\left\\\\{ {\\\\mathbf{x}\\\\,\\\\in \
\\\\mathscr{H}_1 :\\\\,~ T\\\\mathbf{x}=\\\\mathbf{0}} \
\\\\right\\\\}~.\\n\\\\]\\n\\\\end{defn}\\nThe range and null spaces of a \
linear operator and its adjoint are related through the following theorem.\\n\
\\\\newtheorem{AandAPlusSpaces}{Theorem}\\n\\\\begin{thm}\\n\\\\label{\
AandAPlusSpaces}\\nFor a bounded linear operator $T:\\\\mathscr{H}_1 \\\\to \
\\\\mathscr{H}_2 $ whose range space $\\\\mathscr{R}_{T}$ is closed, and \
whose adjoint range space $\\\\mathscr{R}_{T^+}$ is also closed \
\\\\footnote{Although all bounded linear operators between finite dimensional \
spaces have closed ranges, this is not necessarily the case in infinite \
dimensional spaces. The theorem still holds, however, if we replace the \
corresponding range spaces with their closures.}, the following relations \
hold:\\n\\\\begin{subequations}\\n\\\\label{Chapter1-15}\\n\\\\begin{align}\\\
n{\\\\mathscr{R}_{T}}^\\\\bot & = \\\\mathscr{N}_{T^+}, {\\\\mathscr{N}_{T}}^\
\\\\bot = \\\\mathscr{R}_{T^+}, \\\\label{Chapter1-15a} \
\\\\\\\\\\n{\\\\mathscr{R}_{T^+}}^\\\\bot & = \\\\mathscr{N}_{T}, \
{\\\\mathscr{N}_{T^+}}^\\\\bot = \\\\mathscr{R}_{T} \\\\label{Chapter1-15b} \
~,\\n\\\\end{align}\\nwhere the subspaces denoted by the symbol $\\\\bot$ are \
defined in definition \\\\ref{Vperp}. In \
addition,\\n\\\\[\\n\\\\mathscr{H}_1 = \\\\mathscr{R}_{T^ + } \\\\oplus \\\
\\mathscr{N}_T ,~ \\\\mathscr{H}_2 = \\\\mathscr{R}_T \\\\oplus \
\\\\mathscr{N}_{T^ +} ~,\\n\\\\]\\nwhere the direct sum\\\\index{direct sum} \
$\\\\oplus$ is defined in definition \
\\\\ref{InnerAndDirectSum}.\\n\\\\end{subequations}\\n\\\\end{thm}\\n\\nGiven \
a self adjoint (and bounded) positive definite operator $T$, and denoting its \
smallest and largest eigenvalues \\\\footnote{For a positive definite \
operator $T$, the matrix element $\\\\left\\\\langle {x,Tx} \
\\\\right\\\\rangle$ is real and positive for all $x \\\\in \\\\mathscr{V}$. \
The eigenvalues and the eigenfunctions of the operator $T$ are defined by the \
equation $Tx = \\\\lambda x$, where $x$ is an eigenvector (eigenfunction) \
corresponding to the eigenvalue\\\\index{eigenvalue} $\\\\lambda$.} by \
$\\\\lambda _{\\\\min }$ and $\\\\lambda _{\\\\max }$, where $ 0< \\\\lambda \
_{\\\\min } < \\\\lambda _{\\\\max } $, the spectral \
radius\\\\index{spectral radius} of $T$ is defined to be its largest \
eigenvalue $\\\\rho_T \\\\equiv \\\\lambda _{\\\\max }$. Following this, if \
$T$ is not a self adjoint operator we define the $l_2$ norm of $T$ by the \
following \
equation\\n\\\\begin{equation}\\n\\\\label{Chapter1-16}\\n\\\\left\\\\| T \
\\\\right\\\\|_2 = \\\\sqrt {\\\\rho _{T^ + T} } ~ \
.\\n\\\\end{equation}\\nThis definition relies on the fact that the operator \
$T^+ T$ is self adjoint and positive definite for any operator $T$. It is \
self adjoint because $\\\\left( {T^ + T} \\\\right)^ + = T^ + T^{ + + } \
= T^ + T$, and positive definite since $\\\\left\\\\langle {x,T^ + Tx} \
\\\\right\\\\rangle = \\\\left\\\\langle {Tx,Tx} \\\\right\\\\rangle $. The \
latter is real and positive by the postulates of an inner product shown in \
equation \\\\eqref{Chapter1-4}. When $T$ is a self adjoint operator, i.e. \
$T^ + = T$, it follows that $\\\\left\\\\| {T} \\\\right\\\\|_2 = \\\\rho \
_T$.\\n\\\\newtheorem{UnitaryOperator}{Definition}\\n\\\\begin{defn}\\n\\\\\
label{UnitaryOperator}\\nA unitary\\\\index{operator!unitary} operator is one \
that satisfies the equation $T^ + T = 1$, i.e., the operator has a left \
inverse which is the same as its adjoint. \\n\\\\end{defn}\\n\\n\\n\\n\\n\\n\
\\\\section{Reproducing Kernel Hilbert Space\\\\index{reproducing \
kernel!Hilbert space}}\\n\\nA special case of linear operators is a linear \
functional\\\\index{functional} as defined below \
\\\\cite{BaNa66}.\\n\\\\newtheorem{LinearFunctional}{Definition}\\n\\\\begin{\
defn}\\n\\\\label{LinearFunctional}\\nA linear functional on a Hilbert space \
$\\\\mathscr{H}$ is a linear map $f:\\\\mathscr{H} \\\\to \
\\\\mathbb{C}$.\\n\\\\end{defn}\\n\\\\noindent For instance, suppose that $\\\
\\phi \\\\in \\\\mathscr{H}$ is fixed. Then \\n\\\\[\\nf_\\\\phi \\\\left[ \
x \\\\right] = \\\\left\\\\langle {\\\\phi ,x} \\\\right\\\\rangle = \\\\int\
\\\\limits_{ - \\\\infty }^\\\\infty {\\\\phi ^ * \\\\left( t \\\\right)} x\
\\\\left( t \\\\right)dt, \\\\forall x \\\\in \\\\mathscr{H}, \\n\\\\]\\nis \
a linear functional. An important example is the evaluation \
functional\\\\index{functional!evaluation} \
$\\\\mathscr{E}_t$.\\n\\\\newtheorem{EvaluationFunctional}{Definition}\\n\\\\\
begin{defn}\\n\\\\label{EvaluationFunctional}\\nThe evaluation functional $\\\
\\mathscr{E}_t$ is a linear map between a Hilbert space of functions \
$\\\\mathscr{H}$ and the complex numbers $\\\\mathbb{C}$ \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-17}\\n\\\\mathscr{E}_t \
:\\\\mathscr{H} \\\\to \\\\mathbb{C}~, ~~ \\\\mathscr{E}_t [x] = x\\\\left( t \
\\\\right) ~,\\n\\\\end{equation}\\nwhere the functions are assumed to have \
been defined for every point $t \\\\in \
\\\\mathbb{R}$.\\n\\\\end{defn}\\n\\\\newtheorem{BoundedLinearFunctionals}{\
Definition}\\n\\\\begin{defn}\\nA linear functional $f$ on a Hilbert space is \
bounded\\\\index{functional!bounded} if there exists a constant $C$, such \
that $\\\\left| {f\\\\left[ x \\\\right]} \\\\right| \\\\le C\\\\left\\\\| x \
\\\\right\\\\|$, $\\\\forall x$ in the Hilbert space. Note that $C$ is \
independent of $x$.\\n\\\\end{defn}\\n\\\\noindent An example is the Hilbert \
space $L_2 \\\\left( \\\\mathbb{R} \\\\right)$, and the linear functional \\n\
\\\\[\\nf_w \\\\left[ x \\\\right] \\\\equiv \\\\int\\\\limits_{ - \\\\infty \
}^\\\\infty {w^ * \\\\left( t \\\\right)x\\\\left( t \\\\right)dt} \
~.\\n\\\\]\\nThen by the Cauchy-Schwartz inequality \\\\eqref{Chapter1-7} we \
have \\n\\\\[\\n\\\\left| {f_w \\\\left[ x \\\\right]} \\\\right| \\\\le \
\\\\left\\\\| w \\\\right\\\\|\\\\left\\\\| x \\\\right\\\\| ~,\\n\\\\]\\nand \
so we may take $C=\\\\left\\\\| w \
\\\\right\\\\|$.\\n\\\\newtheorem{ContinuousLinearFunctional}{Definition}\\n\\\
\\begin{defn}\\nA linear functional $f$ is \
continuous\\\\index{functional!continuous} if, given that $x_n \\\\left( t \\\
\\right) \\\\to x\\\\left( t \\\\right)$ as $n \\\\to \\\\infty$, then \
$f\\\\left[ {x_n } \\\\right] \\\\to f\\\\left[ x \
\\\\right]$.\\n\\\\end{defn}\\n\\\\newtheorem{\
BoundedContinuousLinearFunctional}{Theorem}\\n\\\\begin{thm}\\n\\\\label{\
BoundedContinuousLinearFunctional}\\nA linear functional $f$ defined on a \
Hilbert space is bounded if, and only if, it is \
continuous.\\n\\\\end{thm}\\nThis theorem, to be used below, can be proven as \
follows. Given that $x_n \\\\to x$ and if $f$ is bounded, we use linearity \
to write \\n\\\\[\\n\\\\left| {f\\\\left( {x_n } \\\\right) - f\\\\left( x \\\
\\right)} \\\\right| = \\\\left| {f\\\\left( {x_n - x} \\\\right)} \
\\\\right| \\\\le C\\\\left\\\\| {x_n - x} \\\\right\\\\| ~, \
\\n\\\\]\\nwhich proves the continuity of the functional. Conversely, if $f$ \
is continuous but unbounded then there must exist a sequence $x_n$ such that \
$f\\\\left[ {x_n } \\\\right] > n\\\\left\\\\| {x_n } \\\\right\\\\|$. Now \
let $x'_n = {{x_n } \\\\mathord{\\\\left/ {\\\\vphantom {{x_n } {\\\\left( \
{n\\\\left\\\\| {x_n } \\\\right\\\\|} \\\\right)}}} \\\\right.\\n \
\\\\kern-\\\\nulldelimiterspace} {\\\\left( {n\\\\left\\\\| {x_n } \
\\\\right\\\\|} \\\\right)}}$. Clearly $\\\\left\\\\| {x'_n } \\\\right\\\\| \
= n^{ - 1}$ and so $x'_n \\\\to 0$. Then $\\\\left| {f\\\\left[ {x'_n } \
\\\\right]} \\\\right| = {{f\\\\left[ {x_n } \\\\right]} \
\\\\mathord{\\\\left/ {\\\\vphantom {{f\\\\left[ {x_n } \\\\right]} \
{\\\\left( {n\\\\left\\\\| {x_n } \\\\right\\\\|} \\\\right) > 1}}} \
\\\\right.\\n \\\\kern-\\\\nulldelimiterspace} {\\\\left( {n\\\\left\\\\| \
{x_n } \\\\right\\\\|} \\\\right) > 1}}$ which shows that $f\\\\left[ {x'_n } \
\\\\right]$ will not tend to zero, even though $x'_n \\\\to 0$, thus \
contradicting the continuity assumption. Consequently if $f$ is continuous \
it must be bounded.\\n\\nIn a Hilbert space, bounded linear functionals have \
a particularly simple representation, embodied in the Frechet-Riesz \
theorem.\\n\\\\newtheorem{FrechetRiesz}{Theorem}\\n\\\\begin{thm}\\n\\\\label{\
FrechetRiesz}\\n(Frechet-Riesz) If $f$ is a bounded linear functional on a \
Hilbert space $\\\\mathscr{H}$, then there exists a unique element $\\\\phi \
\\\\in \\\\mathscr{H}$ such that $f\\\\left[ x \\\\right] = \
\\\\left\\\\langle {\\\\phi ,x} \\\\right\\\\rangle$ for all $x \\\\in \
\\\\mathscr{H}$.\\n\\\\end{thm}\\n\\\\newtheorem{BoundedEvaluationFunctional}{\
defn}\\n\\\\begin{defn}\\n\\\\label{BoundedEvaluationFunctional}\\nThe \
evaluation functional $\\\\mathscr{E}_t$ of definition \
\\\\ref{EvaluationFunctional} is said to be bounded if given $x(t) \\\\in \
\\\\mathscr{H}$, defined for every value of the independent variable $t$, \
there exist constants $C_t$ (that depend on $t$ but not on $x$) for which the \
following inequality holds for all $x$, \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-18}\\n\\\\left| {x\\\\left( t \
\\\\right)} \\\\right| \\\\le C_t \\\\left\\\\| x \
\\\\right\\\\|~.\\n\\\\end{equation}\\n\\\\end{defn}\\nIf the evaluation \
functional is bounded then theorem \\\\ref{FrechetRiesz} shows that \
$\\\\mathscr{E}_t [x]$ can be written as the inner product between a fixed \
element $e_{_t} \\\\in \\\\mathscr{H}$ and $x$, i.e., \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-19}\\nx(t) = \\\\left\\\\langle \
{e_{_t},x} \\\\right\\\\rangle = \\\\int\\\\limits_{ - \\\\infty }^\\\\infty \
{e_{_t}^ * \\\\left( t' \\\\right)x\\\\left( t' \\\\right)dt'} \
~.\\n\\\\end{equation}\\n\\\\newtheorem{ReproducingKernel}{Definition}\\n\\\\\
begin{defn}\\n\\\\label{ReproducingKernel}\\nIn a Hilbert space with a \
bounded evaluation functional the reproducing kernel\\\\index{reproducing \
kernel} is defined by \
\\\\cite{Aron50}\\n\\\\begin{equation}\\n\\\\label{Chapter1-20}\\nK\\\\left( \
{t,t'} \\\\right) \\\\equiv e_{_t} \\\\left(t' \\\\right) = \
\\\\left\\\\langle {e_{t'} ,e_t } \
\\\\right\\\\rangle\\n\\\\end{equation}\\n\\\\end{defn}\\n\\\\noindent \
Equation \\\\eqref{Chapter1-19} can now be written in terms of the \
reproducing \
kernel\\n\\\\begin{equation}\\n\\\\label{Chapter1-21}\\nx\\\\left( t \
\\\\right) = \\\\int\\\\limits_{ - \\\\infty }^\\\\infty {K^* \\\\left( \
{t,t'} \\\\right)} x\\\\left(t' \\\\right)dt' \\\\equiv \\\\left\\\\langle \
{K,x} \\\\right\\\\rangle _t ~, \\n\\\\end{equation}\\nwhere the subscript \
$t$ indicates that in the corresponding inner product the integration is over \
the variable $t'$ and that $t$ is held fixed. This is the reproducing kernel \
property of any Hilbert space in which the evaluation functional is a bounded \
(equivalently, continuous) linear map. Thus, we have the following theorem. \
\\n\\\\newtheorem{Aronszajn}{Theorem}\\n\\\\begin{thm}\\n\\\\label{Aronszajn}\
\\n(Aronszajn\\\\index{Aronszajn}) A necessary and sufficient condition that \
a Hilbert space possesses a reproducing kernel is that the evaluation \
functional be \
bounded.\\n\\\\end{thm}\\n\\\\newtheorem{KernelUniqueness}{Theorem}\\n\\\\\
begin{thm}\\n\\\\label{KernelUniqueness}\\nIf a Hilbert space possesses a \
reproducing kernel then the kernel is unique.\\n\\\\end{thm}\\n\\\\noindent \
For if we have two reproducing kernels $K_1 \\\\left( {t,\\\\tau } \
\\\\right)$ and $K_2 \\\\left( {t,\\\\tau } \\\\right)$, then \
\\n\\\\begin{align}\\n\\\\label{Chapter1-22}\\n\\\\left\\\\| {K_1 \\\\left( \
{t,\\\\tau } \\\\right) - K_2 \\\\left( {t,\\\\tau } \\\\right)} \
\\\\right\\\\|_t ^2 & = \\\\left\\\\langle {K_1 \\\\left( {t,\\\\tau } \
\\\\right) - K_2 \\\\left( {t,\\\\tau } \\\\right),K_1 \\\\left( {t,\\\\tau } \
\\\\right) - K_2 \\\\left( {t,\\\\tau } \\\\right)} \\\\right\\\\rangle _t \
\\\\notag \\\\\\\\ \\n & = \\\\left\\\\langle {K_1 \\\\left( {t,\\\\tau } \\\
\\right) - K_2 \\\\left( {t,\\\\tau } \\\\right),K_1 \\\\left( {t,\\\\tau } \
\\\\right)} \\\\right\\\\rangle _t \\\\notag \\\\\\\\\\n & - \
\\\\left\\\\langle {K_1 \\\\left( {t,\\\\tau } \\\\right) - K_2 \\\\left( {t,\
\\\\tau } \\\\right),K_2 \\\\left( {t,\\\\tau } \\\\right)} \
\\\\right\\\\rangle _t \\\\notag \\\\\\\\ \\n & = K_1 \\\\left( {t,t} \
\\\\right) - K_2 \\\\left( {t,t} \\\\right) - K_1 \\\\left( {t,t} \\\\right) \
+ K_2 \\\\left( {t,t} \\\\right) \\\\notag \\\\\\\\\\n & = 0 \
~.\\n\\\\end{align}\\n\\\\noindent The final result, of course, holds for all \
$t$ and so the reproducing kernel is unique.\\n\\n\\n\\n\\\\section{The Dirac \
Delta Distribution}\\n\\\\label{DiracDelta}\\n\\n\\nIn most Hilbert spaces \
the evaluation functional is unbounded. For instance, functions in \
$L_2(\\\\mathbb{R})$ could be unbounded on sets of measure zero and so a \
reproducing kernel cannot be defined. Examples of reproducing kernel Hilbert \
spaces include the space of band limited functions (a subspace of \
$L_2(\\\\mathbb{R})$ defined in section \\\\ref{BandLimitedSection}) as well \
as the range spaces of both the windowed Fourier transform and the continuous \
wavelet transform (subspaces of $L_2({\\\\mathbb{R}}^2)$), as will be shown \
later. \\n\\nThe Dirac delta\\\\index{Dirac!delta} distribution \
\\\\cite{GeSh64} \\\\footnote{The Dirac delta is popularly known as the Dirac \
delta function. The usual definition ``zero everywhere except at one point\\\
\", however, clearly does not satisfy the basic requirements for a function.} \
$\\\\delta(t-t')$ is a singular form of a reproducing kernel existing in a \
space far larger than $L_2(\\\\mathbb{R})$: the dual space of a space of \
``sufficiently well behaved (good)\\\" \
functions.\\n\\\\newtheorem{TestFunctions}{defn}\\n\\\\begin{defn}\\n\\\\\
label{TestFunctions}\\nConsider the space of functions $\\\\mathscr{S}$ whose \
elements (in general, complex functions) are absolutely integrable on the \
real line $\\\\mathbb{R}$, and are infinitely many times differentiable at \
every $t \\\\in \\\\mathbb{R}$, and, together with any of their derivatives, \
decay to zero faster than any negative power of $t$ as $\\\\left| t \
\\\\right| \\\\to \\\\infty$ \\\\cite{Brem65}. We will call this the space \
of test functions of rapid decay \\\\footnote{Technically, the space of test \
functions $\\\\mathscr{K}$, contained within $\\\\mathscr{S}$, is defined to \
include only functions with continuous derivatives of all order and with \
compact support. Although, $\\\\mathscr{K}$ is used to define generalized \
functions, in practice $\\\\mathscr{S}$ is often used and so we take that as \
our function space for the definition of the Dirac delta and other \
generalized functions.}.\\n\\\\end{defn}\\n\\\\noindent An example of the \
sort of functions contained in $\\\\mathscr{S}$ is $\\\\exp (-t^2)$. \
\\n\\\\newtheorem{TestFunctionsDual}{defn}\\n\\\\begin{defn}\\n\\\\label{\
TestFunctionsDual}\\nThe space of all linear functionals defined on \
$\\\\mathscr{S}$, whose range is either $\\\\mathbb{R}$ or $\\\\mathbb{C}$, \
is known as the dual space of $\\\\mathscr{S}$ and is denoted by \
$\\\\mathscr{S'}$. \\n\\\\end{defn}\\n\\\\noindent Thus, if $\\\\phi_{_1} \
(t) \\\\in \\\\mathscr{S'}$, then $\\\\int\\\\limits_{ - \\\\infty \
}^\\\\infty {\\\\phi_{_1} ^ * \\\\left( t \\\\right)\\\\phi \\\\left( t \
\\\\right) dt}$ is a number (in general, complex). The dual space \
$\\\\mathscr{S'}$ includes all functions in $\\\\mathscr{S}$ as well as all \
functions in $L_2(\\\\mathbb{R})$, and thus it includes regular functions, \
since\\n\\\\[\\n\\\\left| {\\\\int\\\\limits_{- \\\\infty }^\\\\infty {x^*(t) \
\\\\phi(t) dt}} \\\\right| < \\\\infty , ~ x \\\\in L_2(\\\\mathbb{R}) ~ \
{\\\\rm or} ~ x \\\\in \\\\mathscr{S}, ~ \\\\phi \\\\in \
\\\\mathscr{S'}.\\n\\\\]\\nBut there is more: $\\\\mathscr{S'}$ includes \
objects that are not ordinary functions. For instance, consider the linear \
functional that maps every member of $\\\\mathscr{S}$ to the value of that \
member at a fixed point $t'$. This linear map, a member of \
$\\\\mathscr{S'}$, is the Dirac delta generalized function (or, \
distribution). Thus, the defining relation for the Dirac delta \
is\\n\\\\begin{equation}\\n\\\\label{Chapter1-23}\\n\\\\int\\\\limits_{ - \
\\\\infty }^\\\\infty {\\\\delta \\\\left({t-t'} \\\\right)\\\\phi \
\\\\left(t' \\\\right)dt'} = \\\\phi \\\\left( {t} \\\\right) \
~.\\n\\\\end{equation}\\nComparing the above with the reproducing kernel \
property \\\\eqref{Chapter1-21} suggests that the Dirac delta, that is only \
defined on functions in $\\\\mathscr{S}$, is a generalized reproducing \
kernel: it is commonly used with members of $L_2(\\\\mathbb{R})$ which do \
not often even belong to $\\\\mathscr{S}$! Although we can still use \
equation \\\\eqref{Chapter1-23} for functions in $L_2(\\\\mathbb{R})$ at \
points $t$ for which the function is defined, there is no meaning to the \
relation at points $t$ where the function is not defined (comprising a set of \
measure $0$). Thus, the Dirac delta may be considered to be a singular (or \
generalized) reproducing kernel in $L_2(\\\\mathbb{R})$.\\n\\nAnother set of \
useful functions that are members of $\\\\mathscr{S'}$, but not \
$L_2(\\\\mathbb{R})$, are the complex exponentials $e_\\\\omega (t) \\\\equiv \
\\\\left( {2\\\\pi } \\\\right)^{ - {1 \\\\mathord{\\\\left/ {\\\\vphantom {1 \
2}} \\\\right. \\\\kern-\\\\nulldelimiterspace} 2}} \\\\exp (i \\\\omega t)$, \
$\\\\omega \\\\in \\\\mathbb{R}$; they are used to construct the Fourier \
transforms of members of $L_2(\\\\mathbb{R})$. The variable $\\\\omega$ \
(also known as the frequency) is used as a continuous index to distinguish \
different exponential functions. The inverse Fourier transform (the \
reconstruction formula) is based on the \
relation\\n\\\\begin{equation}\\n\\\\label{Chapter1-24}\\n\\\\int\\\\limits_{ \
- \\\\infty }^\\\\infty {e_\\\\omega ^ * \\\\left( t \\\\right)e_\\\\omega \
\\\\left( {t'} \\\\right)d\\\\omega } = \\\\delta \\\\left( {t - t'} \
\\\\right) ~.\\n\\\\end{equation}\\nAs we shall see later, when defining the \
Fourier transform and its inverse, the above equation shows that any member \
of $L_2(\\\\mathbb{R})$ can be expanded in terms of $e_\\\\omega (t)$ \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-25}\\nx(t) = \\\\int\\\\limits_{-\
\\\\infty}^\\\\infty {e_\\\\omega (t) X(\\\\omega)d\\\\omega }, ~~ x(t) \
\\\\in L_2(\\\\mathbb{R}),\\n\\\\end{equation}\\nwhere the expansion \
coefficients $X(\\\\omega)$ \
are\\n\\\\begin{equation}\\n\\\\label{Chapter1-26}\\nX\\\\left( \\\\omega \\\
\\right) = \\\\int\\\\limits_{ - \\\\infty }^\\\\infty {{e_\\\\omega ^*(t)} \
x\\\\left( t \\\\right)dt} ~.\\n\\\\end{equation}\\nThus, the functions $e_\\\
\\omega (t)$ are a continuously labeled basis for members of \
$L_2(\\\\mathbb{R})$ within the space $\\\\mathscr{S'}$. Equation \
\\\\eqref{Chapter1-24} is known as a (generalized ) resolution of \
identity\\\\index{resolution of identity} and often referred to as the \
completeness relation for the continuously indexed ``basis functions\\\" \
$e_\\\\omega (t)$. However, there is no discretization of the frequency \
variable $\\\\omega$ of the form $\\\\omega_n = n\\\\Delta \\\\omega$, $n \
\\\\in \\\\mathbb{Z}$ for which the corresponding functions \
$e_{\\\\omega_{_n}} (t)$ satisfy the completeness relation. We shall see \
later that in order to retain the completeness property using Fourier \
transforms we will have to use window functions of compact support.\\n\\nAs \
will be shown later in section \\\\ref{BandLimitedSection}, if we limit the \
frequency band of the function space $L_2(\\\\mathbb{R})$ to the interval $[-\
\\\\Omega, \\\\Omega]$, $\\\\Omega > 0$, the resulting Hilbert space (space \
of band limited functions\\\\index{band limited functions} that are magnitude \
squared integrable, a subspace of $L_2(\\\\mathbb{R})$) will admit a regular \
reproducing kernel that is the band limited form of the Dirac delta, namely, \
${{\\\\sin \\\\left[ {\\\\Omega \\\\left( {t - t'} \\\\right)} \\\\right]} \\\
\\mathord{\\\\left/ {\\\\vphantom {{\\\\sin \\\\left[ {\\\\pi \\\\left( {t - \
t'} \\\\right)} \\\\right]} {\\\\pi \\\\left( {t - t'} \\\\right)}}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} {\\\\pi \\\\left( {t - t'} \
\\\\right)}}$, in the sense that\\n\\\\[\\n{{\\\\sin \\\\left[ {\\\\Omega \
\\\\left( {t - t'} \\\\right)} \\\\right]} \\\\mathord{\\\\left/ \
{\\\\vphantom {{\\\\sin \\\\left[ {\\\\Omega \\\\left( {t - t'} \\\\right)} \
\\\\right]} {\\\\left[ {\\\\pi \\\\left( {t - t'} \\\\right)} \\\\right]}}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} {\\\\left[ {\\\\pi \\\\left( {t - \
t'} \\\\right)} \\\\right]}} \\\\to \\\\delta ({t-t'}) , ~~ {\\\\rm as} ~ \
\\\\Omega \\\\to \\\\infty ~.\\n\\\\]\\nThis reproducing kernel is a \
regular function (not a distribution) and is often used as an example of a \
Dirac delta convergent sequence. We will return to the Dirac delta and \
reproducing kernels later in the context of frames and orthonormal bases. \
\\n\\n\\n\\n\\n\\\\section[Orthonormal Vectors]{Orthonormal \
Vectors}\\n\\nOrthogonal sets of vectors are particularly important in most \
applications and if, in addition, they are of unit norm then they are \
referred to as orthonormal\\\\index{orthonormal vectors} sets \
\\\\cite{Halm74}.\\n\\\\newtheorem{Orthonormal}{Definition}\\n\\\\begin{defn}\
\\nA set of vectors $\\\\left\\\\{ {\\\\mathbf{x}_j } \\\\right\\\\}$ is said \
to be orthonormal provided $\\\\left\\\\langle {\\\\mathbf{x}_j \
,\\\\mathbf{x}_k} \\\\right\\\\rangle = \\\\delta _{jk} $, where $\\\\delta \
_{jk} $ is the Kroenecker delta\\\\index{Kroenecker delta} \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-27}\\n\\\\delta _{jk} =\\\\left\\\
\\{ \\n{\\\\begin{array}{rl}\\n1 \\\\quad j=k \\\\\\\\ \\n0 \\\\quad j\\\\ne \
k \\\\\\\\ \\n\\\\end{array}} \
\\\\right.\\n\\\\end{equation}\\n\\\\end{defn}\\n\\nGiven any set of vectors \
$\\\\left\\\\{ {\\\\mathbf{x}_1 ,\\\\ldots ,\\\\mathbf{x}_n } \\\\right\\\\}$ \
the Gramm-Schmidt\\\\index{Gramm-Schmidt} method \\\\cite{MaWa64} is used to \
obtain an orthonormal set $\\\\left\\\\{ {\\\\mathbf{e}_1 ,\\\\ldots \
,\\\\mathbf{e}_m } \\\\right\\\\}$ where $m\\\\le n$ and both sets of vectors \
span the same vector space. The method is to choose $\\\\mathbf{e}_1 = {{\\\
\\mathbf{x}_1 } \\\\mathord{\\\\left/ {\\\\vphantom {{\\\\mathbf{x}_1 } \
{\\\\left\\\\| {\\\\mathbf{x}_1 } \\\\right\\\\|}}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {\\\\left\\\\| {\\\\mathbf{x}_1 } \
\\\\right\\\\|}}$ \
and\\n\\\\begin{equation}\\n\\\\label{Chapter1-28}\\n\\\\mathbf{e}'_k \
=\\\\mathbf{x}_k\\n-\\\\sum\\\\limits_{l=1}^{k-1} {\\\\left\\\\langle \
{\\\\mathbf{x}_k ,~\\\\mathbf{e}_l} \\\\right\\\\rangle} \\\\mathbf{e}_l, ~ \
\\\\mathbf{e}_k = \\\\left\\\\| {\\\\mathbf{e'}_k } \\\\right\\\\|^{-1} \
\\\\mathbf{e'}_k, ~ k \\\\ge 2. \\n% \\\\mathbf{e}_k \
=\\\\frac{\\\\mathbf{e}'_k }{\\\\left\\\\| {\\\\mathbf{e}'_k } \
\\\\right\\\\|}, ~ k \\\\ge 2.\\n\\\\end{equation}\\nFor example, if we use \
the space of square integrable real functions on $\\\\left[ {-1,1} \
\\\\right]$ with the inner product $\\\\left\\\\langle{f,g} \
\\\\right\\\\rangle=\\\\int\\\\limits_{-1}^1 {f\\\\left( t \\\\right)} \
g\\\\left( t \\\\right)dt$ , starting with the vectors $\\\\left\\\\{ \
{1,t,t^2,\\\\ldots ,t^n} \\\\right\\\\}$, the Gramm-Schmidt method will \
produce the Legendre polynomials up to degree n. If, on the other hand, we \
use the weighted inner product $\\\\left\\\\langle {f,g} \\\\right\\\\rangle \
= \\\\int\\\\limits_{ - 1}^1 {f\\\\left( t \\\\right)} \\\\left( {1 - t^2 } \
\\\\right)^{ - {1 \\\\mathord{\\\\left/ {\\\\vphantom {1 2}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} 2}} g\\\\left( t \\\\right)dt$, the \
Gramm-Schmidt method will result in the Chebyshev polynomials up to degree n \
(see the exercises).\\n\\n\\n\\n\\\\section[Orthogonal \
Projections]{Orthogonal Projections}\\n\\\\label{OrthogonalProjections}\\n\\n\
\\n\\nGiven a Hilbert space $\\\\mathscr{H}$ and two disjoint subspaces \
$\\\\mathbb{V}$ and $\\\\mathbb{W}$, $\\\\mathbb{V}\\\\cap \
\\\\mathbb{W}=\\\\left\\\\{ {\\\\mathbf{0}} \\\\right\\\\}$, a \
projection\\\\index{projection} onto $\\\\mathbb{V}$ is a mapping $P_v \
:\\\\mathscr{H}\\\\to \\\\mathbb{V}$ so that $P_v \
\\\\mathbf{x}=\\\\mathbf{v}$ where \
$\\\\mathbf{x}=\\\\mathbf{v+w},\\\\;\\\\mathbf{v}\\\\in \
\\\\mathbb{V}\\\\;\\\\mathbf{w}\\\\in \\\\mathbb{W}$. Alternatively, $P_w :\\\
\\mathscr{H}\\\\to \\\\mathbb{W}$ where $P_w \\\\mathbf{x}=\\\\mathbf{w}$. \
Any projection operator $P$ must satisfy the equation $P^2=P$, which can be \
used to define a projection. The Cayley-Hamilton theorem\\\\footnote{The \
Cayley Hamilton theorem states that a square matrix $\\\\mathbf{A}$ whose \
characteristic polynomial is defined by $\\\\chi _\\\\mathbf{A} \\\\left( \
\\\\lambda \\\\right) \\\\equiv \\\\left| {\\\\mathbf{A} - \\\\lambda \
\\\\mathbf{1}} \\\\right|$, satisfies the equation $\\\\chi _\\\\mathbf{A} \\\
\\left( \\\\mathbf{A} \\\\right) = 0$.} can be invoked to state that the \
eigenvalue equation $\\\\lambda ^2=\\\\lambda$ must be the characteristic \
equation for any projection operator; thus, the eigenvalues of a projection \
operator are 0 and 1. \\n\\nIf the projection is defined into the same space, \
i.e. $P:\\\\mathscr{H} \\\\to \\\\mathscr{H}$, then the range and the null \
spaces $\\\\mathscr{R}_P$ and $\\\\mathscr{N}_P$ are disjoint linear \
subspaces of the space $\\\\mathscr{H}$ (the only element common to both is \
the zero vector), and $\\\\mathscr{H}=\\\\mathscr{R}_p +\\\\mathscr{N}_p$, \
i.e. the range and the null spaces are algebraic complements and are not \
necessarily orthogonal. \
\\n\\n\\\\newtheorem{OrthogonalProjection}{Definition}\\n\\\\begin{defn}\\n\\\
\\label{OrthogonalProjection}\\nA projection $P:\\\\mathscr{H}\\\\to \
\\\\mathscr{H}$ is said to be orthogonal\\\\index{projection!orthogonal} if \
its range and null spaces are orthogonal to one another, i.e., \
$\\\\mathscr{R}_P \\\\bot \\\\mathscr{N}_P$ \\\\cite{Luen69}. \
\\n\\\\end{defn}\\n\\\\newtheorem{OrthogonalProjectionTheorem1}{Theorem}\\n\\\
\\begin{thm}\\n\\\\label{OrthogonalProjectionTheorem1}\\nA projection \
$P:\\\\mathscr{H}\\\\to \\\\mathscr{H}$ is orthogonal if, and only if, it is \
self adjoint. \\n\\\\end{thm}\\n\\\\noindent If $P$ is self adjoint, then \
for every $x \\\\in \\\\mathscr{H}$ we write $x \\\\equiv Px + (1 - P)x$ is \
an identity. Now clearly $Px \\\\in \\\\mathscr{R}_p$ and $\\\\left( {1 - P} \
\\\\right)x \\\\in \\\\mathscr{N}_p$, since $P\\\\left( {1 - P} \\\\right)x = \
\\\\left( {P - P^2 } \\\\right)x = 0$ ($P$ being a projection operator \
satisfies $P=P^2$). Any member of $\\\\mathscr{R}_p$ is orthogonal to any \
member of $\\\\mathscr{N}_p$ since \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-29}\\n\\\\left\\\\langle \
{Px,\\\\left( {1 - P} \\\\right)y} \\\\right\\\\rangle = \\\\left\\\\langle \
{x,P^ + \\\\left( {1 - P} \\\\right)y} \\\\right\\\\rangle = \
\\\\left\\\\langle {x,\\\\left( {P - P^2 } \\\\right)y} \\\\right\\\\rangle \
= 0 ~.\\n\\\\end{equation}\\n\\\\newtheorem{OrthogonalProjectionTheorem2}{\
Theorem}\\n\\\\begin{thm}\\n\\\\label{OrthogonalProjectionTheorem2}\\nA \
projection $P:\\\\mathscr{H}\\\\to \\\\mathscr{H}$ is orthogonal if $\\\\left\
\\\\| {Px} \\\\right\\\\| \\\\le \\\\left\\\\| x \\\\right\\\\|$ for every $x \
\\\\in \\\\mathscr{H}$.\\n\\\\end{thm}\\n\\\\noindent To show that \
$\\\\mathscr{R}_p$ is orthogonal to $\\\\mathscr{N}_p$, assume $x \\\\in \
\\\\mathscr{N}^{\\\\bot}_p$ (see definition \\\\ref{Vperp}). Then $y = Px - \
x \\\\in \\\\mathscr{N}_p$ since $Py = P^2 x - Px = Px - Px = 0$. So $Px = x \
+ y$ where $\\\\left\\\\langle {x,y} \\\\right\\\\rangle = 0$. \
Thus,\\n\\\\[\\n\\\\left\\\\| x \\\\right\\\\|^2 \\\\ge \\\\left\\\\| {Px} \
\\\\right\\\\|^2 = \\\\left\\\\| x \\\\right\\\\|^2 + \\\\left\\\\| y \
\\\\right\\\\|^2 \\\\ge \\\\left\\\\| x \\\\right\\\\|^2 ~,\\n\\\\]\\nand \
therefore $y=0$, and $Px=x$, so that $x \\\\in \\\\mathscr{R}_p$. This shows \
that $\\\\mathscr{N}^{\\\\bot}_p \\\\subset \\\\mathscr{R}_p$. Conversely, \
if $x \\\\in \\\\mathscr{R}_p$, then $x=Px$ and we write $x=y+u$ with $y \
\\\\in \\\\mathscr{N}^{\\\\bot}_p$ and $u \\\\in \\\\mathscr{N}_p$. Then, \
$x=Px=Py+Pu=Py=y$, since $y \\\\in \\\\mathscr{N}^{\\\\bot}_p$ and therefore \
$y \\\\in \\\\mathscr{R}_p$. Hence, $x \\\\in \\\\mathscr{N}^{\\\\bot}_p$ so \
that $\\\\mathscr{R}_p \\\\subset \
\\\\mathscr{N}^{\\\\bot}_p$.\\n\\nOrthogonal projections are used to solve \
the problem of approximating a given vector in $\\\\mathscr{H}$ by a vector \
that lies entirely in a linear subspace of $\\\\mathscr{H}$. For instance, \
consider the 3-dimensional Euclidean space $\\\\mathbb{R}^3$, a vector \
$\\\\mathbf{r}=\\\\left( {x,y,z} \\\\right)^T$ where $z\\\\ne 0$, and the \
xy-plane which is a linear subspace of $\\\\mathbb{R}^3$. The problem here is \
to find a vector $\\\\hat{\\\\mathbf{r}}$ in the xy-plane that is the best \
approximation to the original vector $\\\\mathbf{r}$ in the sense that \
$\\\\left\\\\| {\\\\mathbf{r}-\\\\hat {\\\\mathbf{r}}} \\\\right\\\\|$ \
attains its minimum value. The solution is provided by the orthogonal \
projection of the vector $\\\\mathbf{r}$ onto the xy-plane namely, $\\\\hat {\
\\\\mathbf{r}}=\\\\left( {x,y,0} \\\\right)^T$. The error in the \
approximation is given by the magnitude of the error vector $\\\\left( \
{0,0,z} \\\\right)^T$. The latter lies on the z-axis which is the subspace \
orthogonal to the approximating subspace, the xy-plane. This example \
illustrates the orthogonal projection theorem \\\\cite{Luen69, \
Keen88}.\\n\\\\newtheorem{OrthogonalProjectionTheorem3}{Theorem}\\n\\\\begin{\
thm}\\n\\\\label{OrthogonalProjectionTheorem3}\\nGiven a Hilbert space \
$\\\\mathscr{H}$ and a closed linear subspace $\\\\mathbb{V}$ and a point $\\\
\\mathbf{x} \\\\in \\\\mathscr{H}$, there exists a unique point \
$\\\\mathbf{v} \\\\in \\\\mathbb{V}$ that is the orthogonal projection of $\\\
\\mathbf{x}$ onto $\\\\mathbb{V}$ and that minimizes the norm $\\\\left\\\\| \
\\\\mathbf{x - v} \\\\right\\\\|$. In addition, we can write $\\\\mathscr{H} \
= \\\\mathbb{V} \\\\oplus \\\\mathbb{V}^ \\\\bot $, and $\\\\mathbb{V} = \
\\\\mathbb{V}^{ \\\\bot \\\\bot } ~ $.\\n\\\\end{thm}\\n\\n\\\\noindent Note \
that the result $\\\\mathbb{V} \\\\subset \\\\mathbb{V}^ {\\\\bot \\\\bot}$ \
of theorem \\\\ref{VperpSubspace} for an inner product space is now replaced \
by $\\\\mathbb{V} = \\\\mathbb{V}^ {\\\\bot \\\\bot}$ in a Hilbert space. \
The difference is, of course, the completeness property of a Hilbert \
space.\\n\\n\\n\\n\\\\section[Multi Resolution Analysis Subspaces]{Multi \
Resolution Analysis Subspaces}\\n\\\\label{MRASubspaces}\\n\\nAs described in \
the previous section, an orthogonal projection is used in the problem of \
approximating a function using the closest element of a given subspace. An \
important concept in the theory and implementation of the discrete orthogonal \
wavelet transform is the concept of orthogonal projections into nested \
subspaces\\\\index{Nested subspaces} that satisfy a completeness property and \
multi resolution property \\\\cite{Mall89a, Mall99}. \
\\n\\\\newtheorem{Nesting}{Definition}\\n\\\\begin{defn}\\nIn \
$L_2(\\\\mathbb{R})$, a set of closed subspaces $\\\\mathscr{V}_m \\\\subset \
L_2 (\\\\mathbb{R})$, $m \\\\in \\\\mathbb{Z}$, is said to satisfy the \
nesting property if \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-30}\\n\\\\mathscr{V}_m \
\\\\subset \\\\mathscr{V}_{m - 1},~ m \\\\in \\\\mathbb{Z} ~. \
\\n\\\\end{equation}\\n\\\\end{defn}\\n\\\\newtheorem{SuccessiveApproximation}\
{Definition}\\n\\\\begin{defn}\\nA nested set of closed subspaces \
$\\\\mathscr{V}_m$ satisfying the completeness \
properties\\n\\\\begin{equation}\\n\\\\label{Chapter1-31}\\n\\\\bigcup \
\\\\limits_{m \\\\in \\\\mathbb{Z}} \\\\mathscr{V}_m = L_2 \\\\left( \
\\\\mathbb{R} \\\\right),~\\\\bigcap \\\\limits_{m \\\\in \
\\\\mathbb{\\\\mathbb{Z}}} \\\\mathscr{V}_m = \\\\left\\\\{ 0 \\\\right\\\\} \
~,\\n\\\\end{equation}\\nis said to have the successive approximation \
property\\\\index{Nested subspaces!successive approximation}. For such a set \
of nested subspaces we also have\\n\\\\[\\nL_2 (\\\\mathbb{R}) = \\\\mathop \
{Lim}\\\\limits_{m \\\\to - \\\\infty } \\\\mathscr{V}_m , ~~ \\\\left\\\\{ \
\\\\mathbf{0} \\\\right\\\\} = \\\\mathop {Lim}\\\\limits_{m \\\\to \\\\infty \
} \\\\mathscr{V}_m \\n\\\\] \\n\\\\end{defn}\\n\\nIf $L_2( \\\\mathbb{R} )$ \
admits a nested set of subspaces with the successive approximation property \
then any square integrable function can be approximated by an orthogonal \
projection onto any one of the subspaces $\\\\mathscr{V}_m$. The process is \
iterative and is depicted in figure \\\\ref{fig:fig-Chapter1-1}. Note that \
an approximation at stage $m$ produces two results: one is the coarser \
approximation at stage $m+1$, and the other is the detail that is lost \
between the approximations at stage $m$ and $m+1$. The former is an \
orthogonal projection into $\\\\mathscr{V}_{m+1}$, while the latter lies in \
\\n$\\\\mathscr{V}_m^ \
\\\\bot$.\\n\\\\begin{figure}[h!]\\n\\t\\\\begin{centering}\\n\\t\\\\\
includegraphics[width=3in]{./fig-Chapter1-1.pdf}\\n\\t\\\\caption{\
Approximation of a vector using nested subspaces and their orthogonal \
complements.}\\n\\t\\\\label{fig:fig-Chapter1-1}\\n\\t\\\\end{centering}\\n\\\
\\end{figure}\\n\\n\\n\\\\newtheorem{Multiresolution}{Definition}\\n\\\\begin{\
defn}\\n\\\\label{MRASubspacesDefn}\\nA nested set of successive \
approximation subspaces $\\\\mathscr{V}_m$ is said to have the multi \
resolution property\\\\index{Nested subspaces!multi resolution} provided that \
\\\\cite{Daub92}\\n\\\\begin{equation}\\n\\\\label{Chapter1-32}\\nx\\\\left( \
t \\\\right) \\\\in \\\\mathscr{V}_m \\\\Leftrightarrow x\\\\left( {2t} \
\\\\right) \\\\in \\\\mathscr{V}_{m - 1} ~.\\n\\\\end{equation}\\nThe \
subspaces $\\\\mathscr{V}_m$ are then said to form a multi resolution \
analysis set of subspaces and are referred to as a \
MRA\\\\index{MRA}.\\n\\\\end{defn}\\n\\nWe illustrate the concept of a MRA by \
producing a set of nested subspaces of $L_2(\\\\mathbb{R}$ satisfying the \
properties of completeness and multi resolution. Consider the space \
$\\\\mathscr{V}_0$ consisting of all functions that are magnitude square \
integrable and that have constant values in all intervals of length $1$ of \
the form $[n, n+1]$, $n \\\\in \\\\mathbb{Z}$. Similarly, consider the space \
$\\\\mathscr{V}_{-1}$, all functions that are magnitude square integrable and \
that have constant values in all intervals of length ${1 \
\\\\mathord{\\\\left/ {\\\\vphantom {1 2}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} 2}$ of the form $\\\\left[ {{n \
\\\\mathord{\\\\left/ {\\\\vphantom {n 2}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} 2},{n \\\\mathord{\\\\left/ {\\\\vphantom {n \
2}} \\\\right.\\n \\\\kern-\\\\nulldelimiterspace} 2} + {1 \
\\\\mathord{\\\\left/ {\\\\vphantom {1 2}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} 2}} \\\\right]$, in addition to the space \
$\\\\mathscr{V}_{1}$, all functions that are magnitude square integrable and \
that have constant values in all intervals of length $2$ of the form \
$\\\\left[ {2n,2n + 2} \\\\right]$, $n \\\\in \\\\mathbb{Z}$. Three typical \
functions from these three spaces are shown in figure \
\\\\ref{fig:fig-Chapter1-2} \
below.\\n\\\\begin{figure}[h!]\\n\\t\\\\begin{centering}\\n\\t\\\\\
includegraphics[width=5in]{./fig-Chapter1-2.pdf}\\n\\t\\\\caption{Three \
typical functions that are constant on intervals of size $1/2$, $1$ and $2$.}\
\\n\\t\\\\label{fig:fig-Chapter1-2}\\n\\t\\\\end{centering}\\n\\\\end{figure}\
\\nEach one of these spaces is actually a subspace (the zero function is \
common to all of them, and a linear combination of two functions in any one \
space is again a function of the same type). In addition, the spaces are \
nested, i.e., $\\\\mathscr{V}_{ - 1} \\\\supset \\\\mathscr{V}_0 \\\\supset \
\\\\mathscr{V}_1$. For instance, $\\\\mathscr{V}_1 \\\\subset \
\\\\mathscr{V}_0 $, since all functions that are constant on intervals \
$\\\\left[ {2n,2n+2} \\\\right]$, are also constant on intervals $\\\\left[ \
{n,n+1} \\\\right]$, but not vice versa. The multi resolution property, \
easily verified, is interpreted as doubling the approximation resolution in \
going from $\\\\mathscr{V}_0 $ to $\\\\mathscr{V}_{-1}$ in the sense that if \
$x\\\\left( t \\\\right)\\\\in \\\\mathscr{V}_0 $ then $x\\\\left( {2t} \
\\\\right)\\\\in \\\\mathscr{V}_{-1} $. Similarly the resolution is halved \
in going from $\\\\mathscr{V}_0 $ to $\\\\mathscr{V}_1 $ in the sense that if \
$x\\\\left( t \\\\right)\\\\in \\\\mathscr{V}_0 $ then $x\\\\left( {{t \
\\\\mathord{\\\\left/ {\\\\vphantom {t 2}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} 2}} \\\\right) \\\\in \\\\mathscr{V}_1 $. \
Thus, we define the subspaces $\\\\mathscr{V}_m$ of functions that have \
constant values in intervals of length $2^m$, of the form $\\\\left[ {2^m \
n,2^m \\\\left( {n + 1} \\\\right)} \\\\right]$, $m,n \\\\in \\\\mathbb{Z}$ \
and obtain a set of nested subspaces satisfying the completeness and the \
multi resolution properties, also known as the Haar MRA\\\\index{Haar!MRA}. \
The Haar multi resolution approximation to a given function will be seen to \
consist of orthogonal projections onto the subspaces $\\\\mathscr{V}_m$ while \
the Haar wavelet transform coefficients will be the approximation errors, \
i.e., orthogonal transformations onto the orthogonal complement subspaces \
$\\\\mathscr{V}_m ^\\\\bot$.\\n\\nThe key to the fast implementation of the \
discrete wavelet transform is the multi resolution property and the existence \
of complete and orthonormal bases in the subspaces $\\\\mathscr{V}_m$ and $\\\
\\mathscr{V}_m ^\\\\bot$. We will discuss complete and orthonormal bases in \
the next section.\\n\\n\\n\\n\\n\\n\\n\\\\section[Complete and Orthonormal \
Bases]{Complete and Orthonormal Bases in $L_2 \\\\left( \\\\mathbb{R} \
\\\\right)$}\\n\\\\label{CompleteOrthonormalBases}\\n\\n\\nThe concept of an \
orthonormal basis\\\\index{basis!infinite} in a finite dimensional vector \
space can be extended to an infinite dimensional Hilbert space \
$\\\\mathscr{H}$. Later in this chapter we will study the more general \
concept of a frame but for now we define a complete and orthonormal \
basis\\\\index{basis!complete and orthonormal} with respect to which any \
element of a Hilbert space has a unique representation. The concepts of \
decomposition of a function (signal) in terms of a basis set (analysis) and \
its inverse (reconstruction or synthesis) are what transform theory in \
general, and wavelet theory in particular, are based on. For a complete and \
orthonormal basis denoted by $\\\\phi_n$, $n \\\\in \\\\mathbb{Z}$, the \
analysis and synthesis operations are summarized in one \
formula\\n\\\\begin{equation}\\n\\\\label{Chapter1-33}\\nx = \
\\\\sum\\\\limits_n {\\\\left\\\\langle {\\\\phi _n, x} \\\\right\\\\rangle \
\\\\phi _n } ~,\\n\\\\end{equation}\\nwith the inner product representing the \
analysis operation and the entire sum representing the reconstruction or \
synthesis operation. The right hand side is an infinite linear combination \
of the form $\\\\sum\\\\limits_n {c_n \\\\phi _n }$ and so we begin with \
conditions under which infinite sums of an orthonormal set converge \
\\\\cite{Keen88}. \
\\n\\n\\\\newtheorem{SeriesConvergence}{Theorem}\\n\\\\begin{thm}\\n\\\\label{\
SeriesConvergence}\\nFor an orthonormal set $\\\\left\\\\{ {\\\\phi_n ,n \
\\\\in \\\\mathbb{Z}} \\\\right\\\\}$, \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-34}\\n\\\\left\\\\langle \
{\\\\phi_n ,\\\\phi_m } \\\\right\\\\rangle = \\\\delta \
_{nm}~,\\n\\\\end{equation}\\nthe infinite linear combination \
$\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {c_n \\\\phi_n }$ converges \
if, and only if,\\n\\\\[\\n\\\\underline{c} = \\\\left[ { \\\\ldots ,c_n , \\\
\\ldots } \\\\right]^T \\\\in l_2 \\\\left( \\\\mathbb{Z} \\\\right) ~~ \
\\\\Leftrightarrow ~~\\\\sum\\\\limits_{n= - \\\\infty }^\\\\infty {\\\\left| \
{c_n } \\\\right|^2} < \\\\infty ~. \\n\\\\]\\n\\\\end{thm}\\nThe most widely \
used description of completeness of a basis $\\\\phi_n$, $n \\\\in \
\\\\mathbb{Z}$ is that for every $x \\\\in \\\\mathscr{H}$, there is a unique \
expansion \\n\\\\begin{equation}\\n\\\\label{Chapter1-35}\\nx = \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {c_n \\\\phi_n } ~, ~~ c_n \
\\\\in \\\\mathbb{C}~,\\n\\\\end{equation}\\n%with expansion coefficients \
$c_n$ given by \\n%\\\\begin{equation}\\n%\\\\label{Chapter1-35}\\n%c_n \
=\\\\left\\\\langle {\\\\phi_n , x} \\\\right\\\\rangle ~, \
\\n%\\\\end{equation}\\nwhere the equality expressed in equation \
\\\\eqref{Chapter1-35} is not a point-by-point equality, but equality in the \
sense of the convergence in the mean of definition \
\\\\ref{ConvergenceInTheMean}. Thus, \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-36}\\n\\\\left\\\\| {x -\\\\sum\\\
\\limits_{n=-N}^{N} {c_n \\\\;\\\\phi _n } } \\\\right\\\\| \\\\to 0,~~ {\\\
\\rm as} ~ N\\\\to \\\\infty ~.\\n\\\\end{equation}\\nThe following is a more \
formal definition of \
completeness.\\n\\\\newtheorem{CompleteSet}{Definition}\\n\\\\begin{defn}\\\
nAn orthonormal set $\\\\left\\\\{ {\\\\phi_n} \\\\right\\\\}$ in a Hilbert \
space $\\\\mathscr{H}$ is said to be complete\\\\index{basis!complete} if it \
is contained in no other orthonormal set (i.e., it is a maximal orthonormal \
set). \\n\\\\end{defn}\\n\\\\noindent The following theorems show three ways \
to characterize complete and orthonormal sets of \
functions.\\n\\\\newtheorem{CompleteSetTheorem1}{Theorem}\\n\\\\begin{thm}\\n\
\\\\label{CompleteSetTheorem1}\\nAn orthonormal set $\\\\left\\\\{ \
{\\\\phi_n} \\\\right\\\\}$ in a Hilbert space $\\\\mathscr{H}$ is complete \
if, and only if, the only vector that is orthogonal to the set is the zero \
vector $\\\\mathbf{0}$. \
\\n\\\\end{thm}\\n\\\\newtheorem{CompleteSetTheorem2}{Theorem}\\n\\\\begin{\
thm}\\n\\\\label{CompleteSetTheorem2}\\nAn orthonormal set $\\\\left\\\\{ {\\\
\\phi_n} \\\\right\\\\}$ in a Hilbert space $\\\\mathscr{H}$ is complete if \
its closure is the entire space. The closure of the set is defined as the \
set of all infinite sums $\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty \
{c_n } \\\\phi _n$ whose coefficients are in $l_2 \\\\left( \\\\mathbb{Z} \
\\\\right)$, i.e., $\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty \
{\\\\left| {c_n } \\\\right|^2 } < \\\\infty$. \
\\n\\\\end{thm}\\n\\\\newtheorem{CompleteSetTheoremParseval}{Theorem}\\n\\\\\
begin{thm}\\n\\\\label{CompleteSetTheoremParseval}\\nAn orthonormal set \
$\\\\left\\\\{ {\\\\phi_n} \\\\right\\\\}$ in a Hilbert space \
$\\\\mathscr{H}$ is complete if Parseval's relation\\\\index{Parseval's \
relation} holds for every member of the \
space,\\n\\\\begin{equation}\\n\\\\label{Chapter1-37}\\n\\\\left\\\\| x \
\\\\right\\\\|^2=\\\\sum\\\\limits_{n=- \\\\infty}^\\\\infty {\\\\left| \
{\\\\left\\\\langle {\\\\phi_n , x} \\\\right\\\\rangle } \\\\right|^2}, ~~ \
\\\\forall x \\\\in \\\\mathscr{H} ~. \\n\\\\end{equation}\\n\\\\end{thm}\\n\
\\nThe following theorem summarizes the above results \
\\\\cite{Keen88}.\\n\\\\newtheorem{CompleteSetTheorem3}{Theorem}\\n\\\\begin{\
thm}\\n\\\\label{CompleteSetTheorem3}\\nFor an orthonormal set $\\\\left\\\\{ \
{\\\\phi_n ,n \\\\in \\\\mathbb{Z}} \\\\right\\\\}$ in a Hilbert space \
$\\\\mathscr{H}$ the following statements are \
equivalent:\\n\\\\begin{itemize}\\n \\\\item {The set is complete.}\\n \
\\\\item {The only vector that is orthogonal to the set is the zero \
vector.}\\n \\\\item {For every $x$ in $\\\\mathscr{H}$ we have the \
expansion $x = \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty \
{\\\\left\\\\langle {\\\\phi _n ,x} \\\\right\\\\rangle \\\\phi _n }$.}\\n \
\\\\item {The closure of the set is the entire space $\\\\mathscr{H}$.}\\n \
\\\\item {For every $x$ in $\\\\mathscr{H}$, $\\\\left\\\\| x \
\\\\right\\\\|^2 = \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty \
{\\\\left| {\\\\left\\\\langle {\\\\phi _n ,x} \\\\right\\\\rangle } \
\\\\right|^2 }$.}\\n \\\\item {For every $x$ and $y$ in $\\\\mathscr{H}$, \
$\\\\left\\\\langle {x,y} \\\\right\\\\rangle = \\\\sum\\\\limits_{n = - \\\
\\infty }^\\\\infty {\\\\left\\\\langle {x,\\\\phi _n } \\\\right\\\\rangle \
\\\\left\\\\langle {\\\\phi _n ,y} \\\\right\\\\rangle \
}$.}\\n\\\\end{itemize}\\n\\\\end{thm}\\n\\nAn example of an orthogonal set \
that is not complete is the infinite set $\\\\left\\\\{ {\\\\sin \\\\left( \
{nt} \\\\right),\\\\;n\\\\in {\\\\rm Z},\\\\;t\\\\in \\\\left[ {0,2\\\\pi } \
\\\\right]} \\\\right\\\\}$. This set, although orthogonal with respect to \
the usual inner product \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-38}\\n\\\\int\\\\limits_0^{2\\\\\
pi } {\\\\sin \\\\left( {mt} \\\\right)} \\\\sin \\\\left( {nt} \\\\right)dt \
= \\\\pi \\\\delta_{mn} ~,\\n\\\\end{equation}\\nis not complete since any \
member of $\\\\left\\\\{ {\\\\cos \\\\left( {kt} \\\\right),\\\\;k\\\\in \
\\\\mathbb{Z},\\\\;t\\\\in \\\\left[ {0,2\\\\pi } \\\\right]} \\\\right\\\\}$ \
is orthogonal to it.\\n\\nWith respect to a complete and orthonormal basis, \
the expansion coefficients \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-39}\\nc_n = \\\\left\\\\langle {\
\\\\phi _n ,x} \\\\right\\\\rangle = \\\\int\\\\limits_{ - \\\\infty \
}^\\\\infty {\\\\phi _n^ * \\\\left( t \\\\right)} x\\\\left( t \
\\\\right)dt ~,\\n\\\\end{equation}\\nare the transform of a given function \
$x(t)$, and fully describe that function in the sense that the function \
$x(t)$ can be reconstructed from the transform coefficients at almost every \
point $t$, i.e., all points except for a set of measure zero \\\\footnote{The \
Gibbs phenomenon\\\\index{Gibbs phenomenon} is such an example. Given a \
periodic function which is piecewise continuously differentiable with a \
countable number of discontinuities, then its Fourier series expansion \
converges to the value of the function everywhere except at the points of \
discontinuity. If $N>>1$ denotes the number of terms in the Fourier series \
partial sum, then it will overshoot the discontinuity at one end while \
undershooting it at the other end by the same amount, nearly $18\\\\%$.}. In \
other words, the transform relation is uniquely invertible in the form \\n\\\
\\begin{equation}\\n\\\\label{Chapter1-40}\\nx\\\\left( t \\\\right) = \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left\\\\langle {\\\\phi \
_n ,x} \\\\right\\\\rangle } \\\\phi _n \\\\left( t \\\\right) \
~.\\n\\\\end{equation}\\nAronszajn's theorem \\\\ref{Aronszajn} stated that a \
Hilbert space whose evaluation functional is bounded (and therefore \
continuous) admits a reproducing kernel\\\\index{reproducing kernel}. If the \
Hilbert space has a complete orthonormal basis set $\\\\left\\\\{ {\\\\phi_n \
\\\\left( t \\\\right),n \\\\in \\\\mathbb{Z}} \\\\right\\\\}$ then the \
kernel has a simple representation described in the following theorem \
\\\\cite{Davi75}.\\n\\\\newtheorem{BergmanKernel}{Theorem}\\n\\\\begin{thm}\\\
n\\\\label{BergmanKernel}\\nFor a complete and orthonormal basis \
$\\\\left\\\\{ {\\\\phi_n \\\\left( t \\\\right),n \\\\in \\\\mathbb{Z}} \
\\\\right\\\\}$ in a Hilbert space, and for a fixed $t'$, the \
sum\\n\\\\begin{equation}\\n\\\\label{Chapter1-41}\\nK\\\\left( {t ,t'} \
\\\\right) = \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\phi_n^ * \
\\\\left( {t} \\\\right)\\\\phi_n \\\\left( {t'} \\\\right)} \
\\n\\\\end{equation}\\nconverges uniformly and absolutely. Furthermore, the \
function $K$ is the (unique) reproducing kernel for the Hilbert space. \\n\\\
\\end{thm}\\nWe will see an example of this when we discuss the subspace of \
band limited functions within $L_2 \\\\left( \\\\mathbb{R} \\\\right)$ in \
section \\\\ref{BandLimitedSection}. Completeness of an orthonormal system \
can now be stated in terms of the reproducing kernel \
\\\\cite{Davi75}.\\n\\\\newtheorem{CompletenessAndKernel}{Theorem}\\n\\\\\
begin{thm}\\n\\\\label{CompletenessAndKernel}\\nIn a Hilbert space with a \
reproducing kernel $K$, an orthonormal set $\\\\left\\\\{ {\\\\phi_n \
\\\\left( t \\\\right),n \\\\in \\\\mathbb{Z}} \\\\right\\\\}$ is complete \
if, and only if, \\n\\\\begin{equation}\\n\\\\label{Chapter1-42}\\nK\\\\left( \
{t,t} \\\\right) = \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left| \
{\\\\phi _n \\\\left(t \\\\right)} \\\\right|^2 } ~.\\n\\\\end{equation}\\n\\\
\\end{thm}\\n\\n\\n\\n\\n\\n\\\\section{The Dirac Notation}\\n\\n\\nThe Dirac \
notation\\\\index{Dirac!notation} is a compact and useful way to describe \
completeness and orthonormality relations \\\\cite{Dira74}. Although the \
evaluation functional in $L_2 \\\\left( \\\\mathbb{R} \\\\right)$ is not \
bounded and so a proper reproducing Kernel (i.e, square integrable) does not \
exist in that space, a generalized function form of the reproducing Kernel \
can be written using the Dirac delta distribution defined in section \
\\\\ref{DiracDelta},\\n\\\\begin{equation}\\n\\\\label{Chapter1-43}\\nx\\\\\
left( t \\\\right) = \\\\int\\\\limits_{ - \\\\infty }^\\\\infty {\\\\delta \
\\\\left( {t - t'} \\\\right)x\\\\left( {t'} \\\\right)dt'} \
~.\\n\\\\end{equation}\\nThe function $x(t)$ is normally represented by $x$ \
(dropping the $t$ variable) since the value of the function at $t$ is the \
result of using the evaluation functional (see definition \
\\\\ref{EvaluationFunctional}). Dirac introduced the notation $\\\\left| x \
\\\\right\\\\rangle$ and called it a ket vector in the corresponding Hilbert \
space. A generalized evaluation functional along the lines of theorem \
\\\\ref{FrechetRiesz} is then defined so that \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-44}\\nx\\\\left( t \\\\right) = \
\\\\left\\\\langle {t,x} \\\\right\\\\rangle ~,\\n\\\\end{equation}\\nin \
analogy with equation \\\\eqref{Chapter1-19}. Dirac chose the following \
notation for the above inner \
product\\n\\\\begin{equation}\\n\\\\label{Chapter1-45}\\n\\\\left\\\\langle \
{t} \\\\mathrel{\\\\left | {\\\\vphantom {t x}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle ~, \
\\n\\\\end{equation}\\nintroducing the bra vector $\\\\left\\\\langle t \
\\\\right|$. The generalized reproducing kernel, in analogy to the \
definition \\\\eqref{Chapter1-20}, then \
becomes\\n\\\\begin{equation}\\n\\\\label{Chapter1-46}\\n\\\\delta \\\\left( \
{t - t'} \\\\right) \\\\equiv \\\\left\\\\langle {t} \\\\mathrel{\\\\left | {\
\\\\vphantom {t {t'}}} \\\\right. \\\\kern-\\\\nulldelimiterspace} {{t'}} \
\\\\right\\\\rangle ~.\\n\\\\end{equation}\\nEquation \\\\eqref{Chapter1-43} \
then becomes\\n\\\\begin{equation}\\n\\\\label{Chapter1-47}\\n\\\\left\\\\\
langle {t} \\\\mathrel{\\\\left | {\\\\vphantom {t x}} \\\\right. \\\\kern-\\\
\\nulldelimiterspace} {x} \\\\right\\\\rangle = \\\\int\\\\limits_{ - \
\\\\infty }^\\\\infty {dt'\\\\left\\\\langle {t}\\n \\\\mathrel{\\\\left | {\
\\\\vphantom {t {t'}}} \\\\right. \\\\kern-\\\\nulldelimiterspace} {{t'}} \
\\\\right\\\\rangle \\\\left\\\\langle {{t'}} \\\\mathrel{\\\\left | \
{\\\\vphantom {{t'} x}}\\n \\\\right. \\\\kern-\\\\nulldelimiterspace} {x} \\\
\\right\\\\rangle } ~,\\n\\\\end{equation}\\nvalid for all ket vectors \
$\\\\left| x \\\\right\\\\rangle$. Thus, we have the completeness, or \
resolution of identity\\\\index{resolution of identity}, relation \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-48}\\n\\\\int\\\\limits_{ - \
\\\\infty }^\\\\infty {dt'\\\\left| {t'} \\\\right\\\\rangle \
\\\\left\\\\langle {t'} \\\\right|} = \\\\mathbf{1} \
~.\\n\\\\end{equation}\\nIf we further introduce the Einstein integration \
convention that all repeated continuous variables occurring in bras and kets \
are integrated over (we will later introduce the same convention for repeated \
discrete indices, i.e., the Einstein summation convention) we \
have\\n\\\\begin{equation}\\n\\\\label{Chapter1-49}\\n\\\\left| t \\\\right\\\
\\rangle \\\\left\\\\langle t \\\\right| = \\\\mathbf{1} \
~.\\n\\\\end{equation}\\nRepeated continuous variables, such as the variable \
$t$ in the above equation, are also known as dummy variables: we can use \
$t'$ or any other variable in the above equation since an integration over \
the variable is implied.\\n\\nWe now make the following associations, \
defining the ket $\\\\left| {\\\\phi _n } \\\\right\\\\rangle$ and the bra \
$\\\\left\\\\langle {\\\\phi _n } \
\\\\right|$,\\n\\\\begin{equation}\\n\\\\label{Chapter1-50}\\n\\\\phi _n \
\\\\left( t \\\\right) \\\\leftrightarrow \\\\left\\\\langle {t} \\\\mathrel{\
\\\\left | {\\\\vphantom {t {\\\\phi _n }}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace}\\n {{\\\\phi _n }} \\\\right\\\\rangle , ~~~ \
\\\\phi _n^ * \\\\left( t \\\\right) \\\\leftrightarrow \\\\left\\\\langle \
{{\\\\phi _n }} \\\\mathrel{\\\\left | {\\\\vphantom {{\\\\phi _n } t}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace}\\n {t} \\\\right\\\\rangle ~.\\n\
\\\\end{equation}\\nThe generalized reproducing kernel property of equation \
\\\\eqref{Chapter1-46} together with the completeness relation \
\\\\eqref{Chapter1-48}, then lead \
to\\n\\\\begin{equation}\\n\\\\label{Chapter1-51}\\n\\\\left\\\\langle \
{\\\\phi _n ,x} \\\\right\\\\rangle \\\\equiv \\\\int\\\\limits_{ - \
\\\\infty }^\\\\infty {\\\\phi _n^ * \\\\left( t \\\\right)x\\\\left( t \
\\\\right)dt} = \\\\int\\\\limits_{ - \\\\infty }^\\\\infty \
{\\\\left\\\\langle {{\\\\phi _n }} \\\\mathrel{\\\\left | {\\\\vphantom {{\\\
\\phi _n } t}} \\\\right. \\\\kern-\\\\nulldelimiterspace}\\n {t} \\\\right\\\
\\rangle \\\\left\\\\langle {t} \\\\mathrel{\\\\left | {\\\\vphantom {t \
x}}\\n \\\\right. \\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle \
dt} \\\\equiv \\\\left\\\\langle {{\\\\phi _n }}\\n \\\\mathrel{\\\\left | {\
\\\\vphantom {{\\\\phi _n } x}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace}\\n {x} \\\\right\\\\rangle \
~.\\n\\\\end{equation}\\nIn addition to the integration convention we \
introduce the Einstein summation convention: An expression containing a \
repeated index is summed over that index. For instance, using the Dirac \
notation the completeness relation \\\\eqref{Chapter1-40} \
becomes\\n\\\\begin{equation}\\n\\\\label{Chapter1-52}\\n\\\\left| x \
\\\\right\\\\rangle = \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty \
{\\\\left| {\\\\phi _n } \\\\right\\\\rangle } \\\\left\\\\langle {{\\\\phi \
_n }}\\n \\\\mathrel{\\\\left | {\\\\vphantom {{\\\\phi _n } x}}\\n \
\\\\right. \\\\kern-\\\\nulldelimiterspace}\\n {x} \\\\right\\\\rangle ~,\\n\
\\\\end{equation}\\nwhich using the summation convention can be reduced to \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-53}\\n\\\\left| {\\\\phi _n } \
\\\\right\\\\rangle \\\\left\\\\langle {\\\\phi _n } \\\\right| \\\\equiv \
\\\\sum\\\\limits_{n = -\\\\infty }^\\\\infty {\\\\left| {\\\\phi_n } \
\\\\right\\\\rangle \\\\left\\\\langle {\\\\phi_n } \\\\right|}=\\\\mathbf{1} \
~. \\n\\\\end{equation}\\n\\\\noindent Using the Dirac notation the \
orthonormality relation \\\\eqref{Chapter1-34} is now written as \
\\\\footnote{Using the resolution of identity \\\\eqref{Chapter1-55}, \
$\\\\left\\\\langle {{\\\\phi _m }} \\\\mathrel{\\\\left | {\\\\vphantom {{\\\
\\phi _m } {\\\\phi _n }}} \\\\right. \\\\kern-\\\\nulldelimiterspace} \
{{\\\\phi _n }} \\\\right\\\\rangle \\\\equiv \\\\int\\\\limits_{ - \
\\\\infty }^\\\\infty {\\\\left\\\\langle {{\\\\phi _m }} \
\\\\mathrel{\\\\left | {\\\\vphantom {{\\\\phi _m } t}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {t} \\\\right\\\\rangle } \\\\left\\\\langle \
{t} \\\\mathrel{\\\\left | {\\\\vphantom {t {\\\\phi _n }}}\\\\normalfont \
\\\\right. \\\\kern-\\\\nulldelimiterspace} {{\\\\phi _n }} \
\\\\right\\\\rangle dt = \\\\int\\\\limits_{ - \\\\infty }^\\\\infty \
{\\\\phi _m^ * \\\\left( t \\\\right)} \\\\phi _n \\\\left( t \
\\\\right)dt$.}\\n\\\\begin{equation}\\n\\\\label{Chapter1-54}\\n\\\\left\\\\\
langle {{\\\\phi _m }}\\n \\\\mathrel{\\\\left | {\\\\vphantom {{\\\\phi _m } \
{\\\\phi _n }}}\\n \\\\right. \\\\kern-\\\\nulldelimiterspace}\\n {{\\\\phi \
_n }} \\\\right\\\\rangle = \\\\delta _{mn} \
~.\\n\\\\end{equation}\\n\\\\noindent A linear operator acting on a ket \
produces another ket, and together with its bra form we have the relations \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-55}\\nL\\\\left| \\\\phi \
\\\\right\\\\rangle = \\\\left| \\\\psi \\\\right\\\\rangle ~, ~~ \\\\left\\\
\\langle \\\\psi \\\\right| = \\\\left\\\\langle \\\\phi \\\\right|L^ + ~.\
\\n\\\\end{equation}\\nFor instance, the functions $\\\\left( {2\\\\pi } \
\\\\right)^{ - {1 \\\\mathord{\\\\left/ {\\\\vphantom {1 2}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} 2}} \\\\exp \\\\left( {i\\\\omega t} \
\\\\right)$ in the Dirac notation are written as $\\\\left\\\\langle {t} \
\\\\mathrel{\\\\left | {\\\\vphantom {t \\\\omega }} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {\\\\omega } \\\\right\\\\rangle$ and their \
complex conjugates are \\n\\\\[\\n\\\\left\\\\langle {\\\\omega } \
\\\\mathrel{\\\\left | {\\\\vphantom {\\\\omega t}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {t} \\\\right\\\\rangle = \\\\left( \
{2\\\\pi } \\\\right)^{ - {1 \\\\mathord{\\\\left/ {\\\\vphantom {1 2}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} 2}} \\\\exp \\\\left( { - \
i\\\\omega t} \\\\right) ~.\\n\\\\]\\nThus,\\n\\\\[\\n\\\\left\\\\langle \
{{t'}} \\\\mathrel{\\\\left | {\\\\vphantom {{t'} \\\\omega }} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {\\\\omega } \\\\right\\\\rangle \
\\\\left\\\\langle {\\\\omega } \\\\mathrel{\\\\left | {\\\\vphantom \
{\\\\omega t}} \\\\right. \\\\kern-\\\\nulldelimiterspace} {t} \
\\\\right\\\\rangle = \\\\frac{1}{{2\\\\pi }}\\\\int\\\\limits_{ - \\\\infty \
}^\\\\infty {e^{ i\\\\omega \\\\left( {t' - t} \\\\right)} d\\\\omega } = \\\
\\delta \\\\left( {t' - t} \\\\right) = \\\\left\\\\langle {{t'}} \
\\\\mathrel{\\\\left | {\\\\vphantom {{t'} t}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {t} \\\\right\\\\rangle ~, \\n\\\\]\\nusing \
the completeness relation and the integration convention (integration over \
repeated variable $\\\\omega$ is assumed), again leading to the completeness \
relation (resolution of identity\\\\index{resolution of identity}) \
\\n\\\\[\\n\\\\left| \\\\omega \\\\right\\\\rangle \\\\left\\\\langle \
\\\\omega \\\\right| = \\\\mathbf{1} ~. \\n\\\\]\\nThe functions $\\\\left( \
{2\\\\pi } \\\\right)^{ - {1 \\\\mathord{\\\\left/ {\\\\vphantom {1 2}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} 2}} \\\\exp \\\\left( \
{-i\\\\omega t} \\\\right)$ can be thought of as the Fourier basis \
functions\\\\index{Fourier!basis functions} in the expansion of any \
$L_2(\\\\mathbb{R})$ function (excluding all sets of measure $0$ on which \
functions can be badly behaved), but they are not members of \
$L_2(\\\\mathbb{R})$ since they are clearly not square integrable. One way \
to include these functions is by considering the dual space as discussed in \
section \\\\ref{DiracDelta}. Although the existence of the Fourier transform \
will be shown in the next section, this formal point of view has the \
advantage of showing the deficiencies of the continuous Fourier transform and \
motivating the search for other continuous basis function (e.g., the windowed \
Fourier transform and the continuous wavelet transform) whose discretization \
will lead to frames and orthonormal bases.\\n\\n\\n\\n\\\\section{The Fourier \
Transform}\\n\\\\label{Section_FT}\\n\\nThe continuous time Fourier transform\
\\\\index{Fourier!transform} can actually be defined for a larger class of \
functions than $L_2(\\\\mathbb{R})$, namely the class of absolutely \
integrable functions $L_1(\\\\mathbb{R})$, as described in the following \
theorem \\\\cite{Brem65, \
Korn88}.\\n\\\\newtheorem{FTL1}{Theorem}\\n\\\\begin{thm}\\n\\\\label{FTL1}\\\
nLet $x(t) \\\\in L_1(\\\\mathbb{R})$. Then $X(\\\\omega)$ defined \
by\\n\\\\begin{equation}\\n\\\\label{Chapter1-56}\\nX\\\\left( \\\\omega \
\\\\right) = \\\\int\\\\limits_{ - \\\\infty }^\\\\infty {x\\\\left( t \
\\\\right)e^{ - i\\\\omega t} dt} \\n\\\\end{equation}\\nis uniformly \
continuous and bounded for $\\\\omega \\\\in \\\\mathbb{R}$, and $X\\\\left( \
\\\\omega \\\\right) \\\\to 0$ for $\\\\left| \\\\omega \\\\right| \\\\to \
\\\\infty$. In addition, if $X(\\\\omega) \\\\in L_1(\\\\mathbb{R})$, \
then\\n\\\\begin{equation}\\n\\\\label{Chapter1-57}\\nx\\\\left( t \\\\right) \
= \\\\frac{1}{{2\\\\pi }}\\\\int\\\\limits_{ - \\\\infty }^\\\\infty \
{X\\\\left( \\\\omega \\\\right)e^{ + i\\\\omega t} d\\\\omega } \
~,\\n\\\\end{equation}\\nwhere the equality is in the sense of equality in \
the mean (see definition \\\\ref{ConvergenceInTheMean}). Note that \
$X(\\\\omega) \\\\in L_1(\\\\mathbb{R})$ is not implied by $x(t) \\\\in \
L_1(\\\\mathbb{R})$.\\n\\\\end{thm}\\n\\n\\\\noindent A class of functions \
that are not absolutely integrable and still possess a Fourier transform also \
exists, e.g., the function ${{\\\\sin \\\\left( t \\\\right)} \
\\\\mathord{\\\\left/ {\\\\vphantom {{\\\\sin \\\\left( t \\\\right)} t}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} t}$. These functions are covered \
by the following \
theorem.\\n\\\\newtheorem{FTL1-2}{Theorem}\\n\\\\begin{thm}\\n\\\\label{FTL1-\
2}\\nLet $x\\\\left( t \\\\right) = y\\\\left( t \\\\right)\\\\sin \\\\left( \
{\\\\omega t + \\\\phi } \\\\right)$. If $y\\\\left( {t + k} \\\\right) < \
y\\\\left( t \\\\right)$, and if % \\n\\\\[\\n\\\\int\\\\limits_{\\\\left| t \
\\\\right| > \\\\varepsilon > 0} {\\\\left| {{{x\\\\left( t \\\\right)} \
\\\\mathord{\\\\left/ {\\\\vphantom {{x\\\\left( t \\\\right)} t}} \\\\right.\
\\n \\\\kern-\\\\nulldelimiterspace} t}} \\\\right|dt} < \\\\infty \
~,\\n\\\\]\\nthen $X(\\\\omega)$ exists. In addition, the inverse transform \
exists and is given by equation \
\\\\eqref{Chapter1-57}.\\n\\\\end{thm}\\n\\\\noindent For functions in \
$L_2(\\\\mathbb{R})$ we have a stronger \
result.\\n\\\\newtheorem{FTL2}{Theorem}\\n\\\\begin{thm}\\n\\\\label{FTL2}\\n(\
Plancherel) Let $x(t) \\\\in L_2(\\\\mathbb{R})$. Then $X(\\\\omega)$ \
defined by equation \\\\eqref{Chapter1-56} exists. In addition, \
$X(\\\\omega) \\\\in L_2(\\\\mathbb{R})$ and equation \\\\eqref{Chapter1-57} \
holds (in the sense of equality in the mean), i.e., we have \
\\n\\\\[\\nx_\\\\Omega \\\\left( t \\\\right) = \\\\left( {2\\\\pi } \
\\\\right)^{ - 1} \\\\int\\\\limits_{ - \\\\Omega }^\\\\Omega {X\\\\left( \\\
\\omega \\\\right)} e^{i\\\\omega t} d\\\\omega ~, {\\\\rm then} ~~ \
\\\\mathop {Lim}\\\\limits_{\\\\Omega \\\\to \\\\infty } \\\\left\\\\| {x_\\\
\\Omega - x} \\\\right\\\\| = 0 ~.\\n\\\\]\\n\\\\end{thm}\\n\\\\noindent \
The variable $\\\\omega$ denotes the continuous frequency. If we use a \
factor of $\\\\left( {2\\\\pi } \\\\right)^{{{ - 1} \\\\mathord{\\\\left/ {\\\
\\vphantom {{ - 1} 2}} \\\\right. \\\\kern-\\\\nulldelimiterspace} 2}}$ on \
the right hand sides of equations \\\\eqref{Chapter1-56} and \
\\\\eqref{Chapter1-57}, we can write both of the equations in the Dirac \
notation\\n\\\\[\\nX\\\\left( \\\\omega \\\\right) = \\\\left\\\\langle \
{\\\\omega } \\\\mathrel{\\\\left | {\\\\vphantom {\\\\omega t}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {t} \\\\right\\\\rangle \\\\left\\\\langle \
{t}\\n \\\\mathrel{\\\\left | {\\\\vphantom {t x}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle = \
\\\\left\\\\langle {\\\\omega } \\\\mathrel{\\\\left | {\\\\vphantom \
{\\\\omega x}}\\n \\\\right. \\\\kern-\\\\nulldelimiterspace} {x} \
\\\\right\\\\rangle , ~~ x\\\\left( t \\\\right) = \\\\left\\\\langle {t} \
\\\\mathrel{\\\\left | {\\\\vphantom {t \\\\omega }} \\\\right. \
\\\\kern-\\\\nulldelimiterspace}\\n {\\\\omega } \\\\right\\\\rangle \\\\left\
\\\\langle {\\\\omega } \\\\mathrel{\\\\left | {\\\\vphantom {\\\\omega x}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle = \
\\\\left\\\\langle {t}\\n \\\\mathrel{\\\\left | {\\\\vphantom {t x}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle \
~,\\n\\\\]\\nwhich are equivalent to the Fourier reconstruction (inversion) \
formula\\n\\\\[\\n\\\\left| x \\\\right\\\\rangle \\\\equiv \\\\left| \
\\\\omega \\\\right\\\\rangle \\\\left\\\\langle {\\\\omega } \\\\mathrel{\\\
\\left | {\\\\vphantom {\\\\omega x}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace}\\n {x} \\\\right\\\\rangle \
~.\\n\\\\]\\nThus, we adopt the formal view that the vectors $\\\\left| \
\\\\omega \\\\right\\\\rangle$, $\\\\omega \\\\in \\\\mathbb{R}$ form a \
continuously labeled basis satisfying the completeness and orthonormality \
relations\\n\\\\[\\n\\\\left| \\\\omega \\\\right\\\\rangle \
\\\\left\\\\langle \\\\omega \\\\right| = \\\\mathbf{1}, ~~ \
\\\\left\\\\langle {\\\\omega } \\\\mathrel{\\\\left | {\\\\vphantom \
{\\\\omega {\\\\omega '}}}\\n \\\\right. \\\\kern-\\\\nulldelimiterspace} {{\
\\\\omega '}} \\\\right\\\\rangle = \\\\delta \\\\left( {\\\\omega - \
\\\\omega '} \\\\right) ~.\\n\\\\]\\nThe orthonormality relation follows \
easily by inserting the resolution of identity\\\\index{resolution of \
identity} \\\\eqref{Chapter1-48}, or \\\\eqref{Chapter1-49}, in between the \
inner product, i.e., \\n\\\\[\\n\\\\left\\\\langle {\\\\omega }\\n \
\\\\mathrel{\\\\left | {\\\\vphantom {\\\\omega {\\\\omega '}}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {{\\\\omega '}} \\\\right\\\\rangle = \
\\\\left\\\\langle {\\\\omega } \\\\mathrel{\\\\left | {\\\\vphantom \
{\\\\omega t}}\\n \\\\right. \\\\kern-\\\\nulldelimiterspace} {t} \
\\\\right\\\\rangle \\\\left\\\\langle {t} \\\\mathrel{\\\\left | \
{\\\\vphantom {t {\\\\omega '}}} \\\\right. \\\\kern-\\\\nulldelimiterspace} \
{{\\\\omega '}} \\\\right\\\\rangle = \\\\left( {2\\\\pi } \\\\right)^{ - 1} \
\\\\int\\\\limits_{ - \\\\infty }^\\\\infty {e^{ - it\\\\left( {\\\\omega - \
\\\\omega '} \\\\right)} dt} = \\\\delta \\\\left( {\\\\omega - \\\\omega \
'} \\\\right)~.\\n\\\\]\\nWe also adopt the convention of using equations \
\\\\eqref{Chapter1-56} and \\\\eqref{Chapter1-57} as the definition of the \
forward and the inverse Fourier transform instead of their definition in \
terms of the normalized basis functions $\\\\left\\\\langle {\\\\omega } \
\\\\mathrel{\\\\left | {\\\\vphantom {\\\\omega t}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {t} \\\\right\\\\rangle$ and their complex \
conjugates, which include a factor of $\\\\left( {2\\\\pi } \\\\right)^{ - {1 \
\\\\mathord{\\\\left/ {\\\\vphantom {1 2}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} 2}}$. With this convention the equality of \
the energy in the time and the frequency domain known as the Parseval's \
relation\\\\index{Parseval's relation} \\\\footnote{If we include the factors \
of $\\\\left( {2\\\\pi } \\\\right)^{{{ - 1} \\\\mathord{\\\\left/ \
{\\\\vphantom {{ - 1} 2}} \\\\right. \\\\kern-\\\\nulldelimiterspace} 2}}$ on \
the right hand sides of equations \\\\eqref{Chapter1-56} and \
\\\\eqref{Chapter1-57}, then the $2 \\\\pi$ in the last integral of \
Parseval's relation \\\\eqref{Chapter1-58} must be omitted.}, i.e., \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-58}\\n\\\\left\\\\| x \\\\right\\\
\\|^2\\\\equiv \\\\int\\\\limits_{-\\\\infty }^\\\\infty {\\\\left| {x(t)} \\\
\\right|^2 dt}=\\\\frac{1}{2\\\\pi}\\\\int\\\\limits_{-\\\\infty }^\\\\infty \
{\\\\left| {X(\\\\omega)} \\\\right|^2 d\\\\omega } \
~,\\n\\\\end{equation}\\nfollows from the following \
theorem.\\n\\\\newtheorem{FTParseval}{Theorem}\\n\\\\begin{thm}\\n\\\\label{\
FTParseval}\\nLet $x(t), y(t) \\\\in L_2(\\\\mathbb{R})$. Denoting their \
Fourier transforms by $X(\\\\omega)$ and $Y(\\\\omega)$ we \
have\\n\\\\begin{equation}\\n\\\\label{Chapter1-59}\\n\\\\int\\\\limits_{-\\\\\
infty }^\\\\infty {x^\\\\ast \\\\left( {t-k} \\\\right)y\\\\left( {t-l} \
\\\\right)} \\\\;dt=\\\\frac{1}{2\\\\pi }\\\\int\\\\limits_{-\\\\infty \
}^\\\\infty {X^\\\\ast \\\\left( \\\\omega \\\\right)e^{i\\\\omega k}\\\\;Y\\\
\\left( \\\\omega \\\\right)e^{-i\\\\omega l}} \\\\;d\\\\omega \
~.\\n\\\\end{equation}\\n\\\\end{thm}\\n\\\\noindent This can be proven using \
\\n\\\\[\\nx\\\\left( {t-k} \\\\right)=\\\\frac{1}{2\\\\pi \
}\\\\int\\\\limits_{-\\\\infty }^\\\\infty {X\\\\left( \\\\omega \
\\\\right)e^{i\\\\omega \\\\left( {t-k} \\\\right)}\\\\;d\\\\omega } \
~.\\n\\\\]\\n\\nThe rate of decay of the Fourier transform $X(\\\\omega)$ (as \
$\\\\omega \\\\to \\\\infty$) is a measure of the smoothness of the function \
$x(t)$: the faster the decay, the smoother the function. Similarly, the rate \
of decay of $x(t)$ is determined by the smoothness of $X(\\\\omega)$. The \
following three theorems establish these results \\\\cite{Brem65, Korn88}.\\n\
\\\\begin{thm}\\n\\\\label{FTDecay1}\\nGiven the function $x(t)\\\\in \
L_2(\\\\mathbb{R})$ assume that its Fourier transform $X(\\\\omega)$ is $N$ \
times continuously differentiable and all derivatives tend to $0$ as \
$\\\\left| \\\\omega \\\\right| \\\\to \\\\infty$. Then $t^N x\\\\left( t \
\\\\right) \\\\to 0$ as $t \\\\to \\\\infty$.\\n\\\\end{thm}\\n\\\\begin{thm}\
\\n\\\\label{FTDecay2}\\nGiven the function $x(t)\\\\in L_2(\\\\mathbb{R})$ \
assume that its Fourier transform $X(\\\\omega)$ decays sufficiently fast so \
that $\\\\omega ^N X\\\\left( \\\\omega \\\\right)$ is absolutely integrable \
for some positive integer $N$. Then $x(t)$ is $N$ times continuously \
differentiable and \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-60}\\n\\\\frac{{d^n }}{{dt^n \
}}x\\\\left( t \\\\right) = \\\\frac{1}{{2\\\\pi }}\\\\int\\\\limits_{ - \
\\\\infty }^\\\\infty {\\\\left( {i\\\\omega } \\\\right)^n X\\\\left( \
\\\\omega \\\\right)e^{i\\\\omega t} d\\\\omega }, ~~ 1 \\\\le n \\\\le \
N.\\n\\\\end{equation}\\n\\\\end{thm}\\n\\\\begin{thm}\\n\\\\label{FTDecay3}\\\
nGiven the function $x(t)\\\\in L_2(\\\\mathbb{R})$ and its Fourier transform \
$X(\\\\omega)$ assume that $x(t)$ decays sufficiently fast so that $t^N x(t)$ \
is absolutely integrable for some positive integer $N$. Then $X(\\\\omega)$ \
is $N$ times continuously differentiable and \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-61}\\n\\\\frac{{d^n \
}}{{d\\\\omega ^n }}X\\\\left( \\\\omega \\\\right) = \\\\frac{1}{{2\\\\pi \
}}\\\\int\\\\limits_{ - \\\\infty }^\\\\infty {\\\\left( { - it} \
\\\\right)^n x\\\\left( t \\\\right)e^{ - i\\\\omega t} dt}, ~~1 \\\\le n \
\\\\le N.\\n\\\\end{equation}\\n\\\\end{thm}\\n\\n\\nAn important shortcoming \
of the continuously labeled vectors $\\\\left| \\\\omega \
\\\\right\\\\rangle$, $\\\\omega \\\\in \\\\mathbb{R}$, is the absence of a \
discretization of the frequency variable that would produce a frame (see \
section \\\\ref{FramesInInfinite}) let alone an orthonormal basis. The issue \
can be resolved only if we construct continuously labeled functions of $2$ \
variables by using time domain window functions with the vectors $\\\\left| \
\\\\omega \\\\right\\\\rangle$; a procedure known as the windowed Fourier \
transform. If, on the other hand, we limit our Hilbert space to functions of \
compact support then the resulting Fourier transform, when discretized \
appropriately, leads to an orthonormal basis and the transform is known as \
the Fourier series expansion\\\\index{Fourier!series expansion}. \\n\\n\\n\\n\
\\\\section{The Fourier Series Expansion}\\n\\\\label{FourierSeriesExpansion}\
\\n\\n\\nThe space of square integrable functions with support in the \
interval $\\\\left[ {0,T_0} \\\\right]$ denoted by $L_2 \\\\left[ {0,T_0 } \\\
\\right]$ is an example of a Hilbert space with an orthonormal basis that \
results from a discretization of the exponential functions $\\\\exp \\\\left( \
{i\\\\omega t} \\\\right)$ at frequencies $\\\\omega _n = n\\\\omega _0$ \
\\\\cite{Davi63}. Thus, the set of \
functions\\n\\\\begin{equation}\\n\\\\label{Chapter1-62}\\n\\\\phi_n(t) = \
{T_0}^{ - {1 \\\\mathord{\\\\left/ {\\\\vphantom {1 2}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} 2}} \\\\exp \\\\left( {in\\\\omega _0 t} \
\\\\right), ~~ n \\\\in \\\\mathbb{Z},\\n\\\\end{equation}\\nis complete (see \
theorem \\\\ref{Riesz-Fischer} below) and orthonormal \\\\footnote{when \
proving orthonormality we use the result $\\\\mathop \
{Lim}\\\\limits_{\\\\varepsilon \\\\to 0} \\\\left[ {{{\\\\left( {1 - e^{ - \
i\\\\omega _0 {T_0}\\\\varepsilon }} \\\\right)} \\\\mathord{\\\\left/ \
{\\\\vphantom {{\\\\left( {1 - e^{ - i\\\\omega _0 {T_0}\\\\varepsilon } } \\\
\\right)} {i\\\\omega _0 \\\\varepsilon }}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {i\\\\omega _0 \\\\varepsilon }}} \\\\right] \
= T_0$} on the given interval provided that $\\\\omega _0 T_0 = \
2\\\\pi$,\\n\\\\begin{equation}\\n\\\\label{Chapter1-63}\\nT_0^{-1} \\\\int\\\
\\limits_0^{T_0} {e^{ - im\\\\omega _0 t} e^{in\\\\omega _0 t} dt} = \
\\\\delta _{mn}, ~~\\\\omega _0 T_0 = 2\\\\pi ~.\\n\\\\end{equation}\\nUsing \
the Dirac notation and the summation convention we have\\n\\\\begin{equation}\
\\n\\\\label{Chapter1-64}\\n\\\\left\\\\langle {t} \\\\mathrel{\\\\left | {\\\
\\vphantom {t n}} \\\\right. \\\\kern-\\\\nulldelimiterspace} {n} \\\\right\\\
\\rangle \\\\equiv \\\\phi _n \\\\left( t \\\\right) ~, ~~ \
\\\\left\\\\langle {m} \\\\mathrel{\\\\left | {\\\\vphantom {m n}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {n} \\\\right\\\\rangle=\\\\delta_{mn} , ~~ \
\\\\left| n \\\\right\\\\rangle \\\\left\\\\langle n \\\\right|=\\\\mathbf{1} \
~,\\n\\\\end{equation}\\nwhere the requirement $\\\\omega _0 T_0 = 2\\\\pi$ \
is understood throughout this section. A function $x(t)=\\\\left\\\\langle \
{t} \\\\mathrel{\\\\left | {\\\\vphantom {t x}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle$ in this space has \
the Fourier series \
expansion\\n\\\\begin{equation}\\n\\\\label{Chapter1-65}\\n\\\\left| x \
\\\\right\\\\rangle = c_n \\\\left| n \\\\right\\\\rangle ,c_n \\\\equiv \\\
\\left\\\\langle {n} \\\\mathrel{\\\\left | {\\\\vphantom {n x}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace}\\n {x} \\\\right\\\\rangle \
~.\\n\\\\end{equation}\\nParseval's relation\\\\index{Parseval's relation} is \
now \\n\\\\begin{equation}\\n\\\\label{Chapter1-66}\\n\\\\left\\\\| x \
\\\\right\\\\|^2 = \\\\int\\\\limits_0^{T_0} {\\\\left| {x\\\\left( t \
\\\\right)} \\\\right|^2 dt} = \\\\sum\\\\limits_{n = - \\\\infty \
}^\\\\infty {\\\\left| {c_n } \\\\right|^2 } ~.\\n\\\\end{equation}\\nThus, \
the transform coefficients $c_n$ have finite energy. The space of all such \
sequences is the Hilbert space $l_2(\\\\mathbb{Z})$. Starting with a \
sequence $\\\\left\\\\{ {c_n ,n \\\\in \\\\mathbb{Z}} \\\\right\\\\}$ in the \
Hilbert space $l_2 \\\\left( \\\\mathbb{Z} \\\\right)$ the Riesz-Fischer \
theorem\\\\index{Riesz-Fischer theorem} ensures the existence of a function \
$x(t)$ defined on the interval $\\\\left[ {0,T_0 } \\\\right]$ and whose \
Fourier series coefficients are the given sequence $c_n$.\\n\\\\begin{thm}\\n\
\\\\label{Riesz-Fischer}\\n(Riesz-Fischer) Given a sequence $\\\\left\\\\{ \
{c_n ,n \\\\in \\\\mathbb{Z}} \\\\right\\\\}$ in the Hilbert space $l_2 \
\\\\left( \\\\mathbb{Z} \\\\right)$ and the functions $\\\\phi_n(t)$ defined \
in \\\\eqref{Chapter1-62} together with the constraint $\\\\omega _0 T_0 = \
2\\\\pi$, there exists a (Lebesgue) measurable function $x(t)$ defined on the \
interval $\\\\left[ {0,T_0} \\\\right]$ such that\\n\\\\[\\n\\\\mathop \
{Lim}\\\\limits_{N \\\\to \\\\infty } \\\\int\\\\limits_0^{T_0 } {\\\\left| \
{x\\\\left( t \\\\right) - \\\\sum\\\\limits_{n = - N}^N {c_n \\\\phi _n \\\
\\left( t \\\\right)}} \\\\right|} ^2 dt = 0 ~.\\n\\\\]\\nIn addition, $c_n \
= \\\\left\\\\langle {\\\\phi _n ,x} \\\\right\\\\rangle $ and Parseval's \
relation, equation \\\\eqref{Chapter1-66}, holds \
\\\\cite{Beal04}.\\n\\\\end{thm}\\nIt is common to use \\n\\\\begin{equation}\
\\n\\\\label{Chapter1-67}\\nx\\\\left( t \\\\right) = {T_0}^{-1} \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {c_n } e^{ + in\\\\omega _0 \
t}, \\n\\\\end{equation}\\nas the Fourier reconstruction formula. The Fourier \
series coefficients are then given \
by\\n\\\\begin{equation}\\n\\\\label{Chapter1-68}\\nc_n = \
\\\\int\\\\limits_0^{T_0} {e^{ - in\\\\omega _0 t} x\\\\left( t \\\\right)dt} \
,\\n\\\\end{equation}\\nomitting the $T_0^{-{1 \\\\mathord{\\\\left/ \
{\\\\vphantom {1 2}} \\\\right. \\\\kern-\\\\nulldelimiterspace} 2}}$ factor \
that would be present in $\\\\left\\\\langle {n} \\\\mathrel{\\\\left | \
{\\\\vphantom {n x}}\\n \\\\right. \\\\kern-\\\\nulldelimiterspace} {x} \
\\\\right\\\\rangle$, while Parseval's relation becomes\\n\\\\begin{equation}\
\\n\\\\label{Chapter1-69}\\n\\\\left\\\\| x \\\\right\\\\|^2 = \
\\\\int\\\\limits_0^ {T_0} {\\\\left| {x\\\\left( t \\\\right)} \\\\right|^2 \
dt} = {T_0}^{-1} \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left| \
{c_n } \\\\right|^2 } ~,\\n\\\\end{equation}\\nwhich is a special case of the \
more general \
equation\\n\\\\begin{equation}\\n\\\\label{Chapter1-70}\\n\\\\int\\\\limits_0^\
{T_0} {x^ * \\\\left( {t - \\\\tau _1 } \\\\right)y\\\\left( {t - \\\\tau _2 \
} \\\\right)dt} = {T_0}^{-1} \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty \
{c_n^ * } d_n e^{ + in\\\\omega _0 \\\\left( {\\\\tau _1 - \\\\tau _2 } \\\
\\right)} ,\\n\\\\end{equation}\\nwith $c_n$ and $d_n$ denoting the Fourier \
series coefficients of two function $x(t)$ and \
$y(t)$.\\n\\n\\n\\n\\n\\n\\\\section[The Discrete Time Fourier Transform]{The \
Discrete Time Fourier Transform}\\n\\nIf we consider the Hilbert space of \
discrete time sequences with finite energy $l_2(\\\\mathbb{Z}) $, i.e., \
$\\\\left\\\\{ {s\\\\left[ n \\\\right]:\\\\;\\\\sum\\\\limits_{j=-\\\\infty \
}^\\\\infty {\\\\left| {s\\\\left[ n \\\\right]} \\\\right|^2<\\\\infty } } \
\\\\right\\\\}$, then we have the discrete time Fourier transform \
coefficients \
\\\\cite{OpSc75}\\n\\\\begin{equation}\\n\\\\label{Chapter1-71}\\nS\\\\left( \
\\\\omega \\\\right)=\\\\sum\\\\limits_{n=-\\\\infty }^\\\\infty {s\\\\left[ \
n \\\\right]} ~ e^{-in\\\\omega }\\n\\\\end{equation}\\nwhere \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-72}\\ns\\\\left[ n \
\\\\right]=\\\\frac{1}{2\\\\pi \\n}\\\\int\\\\limits_{-\\\\pi }^\\\\pi \
{S\\\\left( \\\\omega \\\\right)\\\\,} e^{in\\\\omega }d\\\\omega \
\\n\\\\end{equation}\\nClearly the Fourier transform $S\\\\left( \\\\omega \\\
\\right)$ is periodic in the frequency variable $\\\\omega $ with a period of \
$2\\\\pi$. Consequently, the inversion formula is an integral over any \
interval of length $2\\\\pi $ which, without loss of generality, we take to \
be the interval $\\\\left[ {-\\\\pi ,\\\\pi } \\\\right]$ \\\\footnote{A \
discrete time sequence $s\\\\left[ n \\\\right]$ is usually the result of \
sampling of a continuous time function $s\\\\left( t \\\\right)$ at times $t \
= n \\\\Delta T = {n \\\\mathord{\\\\left/ {\\\\vphantom {n {f_s }}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} {f_s }}$. The exponential \
function in the Fourier transform is then of the form $e^{ - i{{n\\\\omega } \
\\\\mathord{\\\\left/ {\\\\vphantom {{n\\\\omega } {f_s }}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {f_s }}}$ where now $\\\\omega \\\\in \
\\\\left[ { - \\\\pi f_s , + \\\\pi f_s ,} \\\\right]$. It is common \
practice, without loss of generality, to take the sampling frequency in this \
case to be equal to $1$ Hz, and use a normalized frequency variable \
$\\\\omega$ in the range $\\\\left[ { - \\\\pi , + \\\\pi } \\\\right]$.}. \
The periodicity in $\\\\omega $ now leads to the following relation (a \
special case of the Poisson summation formula \\\\footnote{The Poisson \
summation formula is $ 2\\\\pi \\\\sum\\\\limits_{n = - \\\\infty \
}^\\\\infty s \\\\left( {t + 2n\\\\pi } \\\\right) = \\\\sum\\\\limits_{k = \
- \\\\infty }^\\\\infty {S\\\\left( k \\\\right)} e^{ikt}$, where \
$S\\\\left( \\\\omega \\\\right)$ is the Fourier transform of $s\\\\left( t \
\\\\right)$.})\\n\\\\begin{equation}\\n\\\\label{Chapter1-73}\\n2\\\\pi \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {e^{in\\\\left( {\\\\omega \
- \\\\omega '} \\\\right)} } = \\\\sum\\\\limits_{k = - \\\\infty \
}^\\\\infty {\\\\delta \\\\left( {\\\\omega - \\\\omega ' + 2k\\\\pi } \
\\\\right)} \\\\end{equation}\\nwhich can be used to derive Parseval's \
relation\\\\index{Parseval's relation} in the form \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-74}\\n\\\\sum\\\\limits_{n=-\\\\\
infty }^\\\\infty {\\\\left| {s\\\\left[ n \\\\right]} \\\\right|^2=} \
\\\\frac{1}{2\\\\pi }\\\\int\\\\limits_{-\\\\pi }^\\\\pi {\\\\left| \
{S\\\\left( \\\\omega \\\\right)} \\\\right|^2\\\\,} \
d\\\\omega\\n\\\\end{equation}\\nor in the more general \
form\\n\\\\begin{equation}\\n\\\\label{Chapter1-75}\\n\\\\sum\\\\limits_{n=-\\\
\\infty }^\\\\infty {s^\\\\ast \\\\left[ {n-k} \\\\right]q\\\\left[ {n-l} \
\\\\right]=} \\\\frac{1}{2\\\\pi }\\\\int\\\\limits_{-\\\\pi }^\\\\pi \
{S^\\\\ast \\\\left( \\\\omega \\n\\\\right)\\\\,} e^{-i\\\\omega \
k}Q\\\\left( \\\\omega \\\\right)e^{i\\\\omega l}d\\\\omega \
~.\\n\\\\end{equation}\\n\\n\\n\\n\\n\\\\section[The Discrete Fourier \
Transform]{The Discrete Fourier Transform}\\n\\nThe space of discrete time \
sequences of finite energy that have a finite length $N$ is a finite \
dimensional vector space of dimension $N$. An orthonormal basis for this \
space is provided by the functions \\\\cite{OpSc75} \\n\\\\begin{equation}\\n\
\\\\label{Chapter1-76}\\np_{_{kn}} = N^{{{ - 1} \\\\mathord{\\\\left/ \
{\\\\vphantom {{ - 1} 2}} \\\\right. \\\\kern-\\\\nulldelimiterspace} 2}} \
\\\\exp \\\\left( {{{2i\\\\pi kn} \\\\mathord{\\\\left/ {\\\\vphantom \
{{2i\\\\pi kn} N}} \\\\right. \\\\kern-\\\\nulldelimiterspace} N}} \
\\\\right), ~ ~ \\\\sum\\\\limits_{k = 0}^{N - 1} {p_{_{mk}}^ * } p_{_{kn}} \
= \\\\delta _{mn} ~.\\n\\\\end{equation}\\nFor $x[n], ~ n=0, \\\\ldots, N-1$ \
in this vector space we have the discrete and finite length Fourier series \
representation \\n\\\\begin{equation}\\n\\\\label{Chapter1-77}\\nx\\\\left[ n \
\\\\right]=\\\\sum\\\\limits_{k=0}^{N-1} {X\\\\left[ k \\\\right]\\\\,p_{_n} \
\\\\left[ k \\\\right]} \\\\equiv \\\\frac{1}{\\\\sqrt N \
}\\\\sum\\\\limits_{k=0}^{N-1} {X\\\\left[ k \\\\right]\\\\,} \
e^{\\\\frac{2i\\\\pi kn}{N}},\\n\\\\end{equation}\\nwhere the coefficients \
are given by \\n\\\\begin{equation}\\n\\\\label{Chapter1-78}\\nX\\\\left[ k \
\\\\right]=\\\\frac{1}{\\\\sqrt N }\\\\sum\\\\limits_{n=0}^{N-1} {x\\\\left[ \
n \\\\right]\\\\,} e^{-\\\\,\\\\frac{2i\\\\pi kn}{N}} \
~.\\n\\\\end{equation}\\nParseval's relation is \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-79}\\n\\\\sum\\\\limits_{k=0}^{N-\
1} {\\\\left| {X\\\\left[ k \\\\right]} \\\\right|^2\\\\,} \
=\\\\sum\\\\limits_{n=0}^{N-1} {\\\\left| {x\\\\left[ n \\\\right]} \
\\\\right|^2\\\\,}\\n\\\\end{equation}\\n\\nThe above equations define the \
Discrete Fourier Transform (DFT), that is often implemented by (and referred \
to as) the Fast Fourier Transform (FFT). A more conventional definition for \
the transform pair $x\\\\left[ n \\\\right] \\\\leftrightarrow X\\\\left[ k \
\\\\right]$ is given by \
\\\\cite{OpSc75}\\n\\\\begin{equation}\\n\\\\label{Chapter1-80}\\nX\\\\left[ \
k \\\\right]=\\\\sum\\\\limits_{n=0}^{N-1} {x\\\\left[ n \\\\right]\\\\,} \
e^{-\\\\,\\\\frac{2i\\\\pi kn}{N}},\\\\;x\\\\left[ n \
\\\\right]=\\\\frac{1}{N}\\\\sum\\\\limits_{k=0}^{N-1} {X\\\\left[ k \
\\\\right]\\\\,} \\ne^{\\\\frac{2i\\\\pi kn}{N}}\\n\\\\end{equation}\\nThe \
difference with the original definition is in the normalization of the \
transform pair. Parseval's relation is now \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-81}\\n\\\\sum\\\\limits_{k=0}^{N-\
1} {\\\\left| {X\\\\left[ k \\\\right]} \\\\right|^2\\\\,} \
=\\\\frac{1}{N}\\\\sum\\\\limits_{n=0}^{N-1} {\\\\left| {x\\\\left[ n \
\\\\right]} \\\\right|^2\\\\,} \
\\n\\\\end{equation}\\n\\n\\n\\n\\\\section[Band Limited Functions]{Band \
limited functions and the sampling \
theorem}\\n\\\\label{BandLimitedSection}\\n\\nA band limited\\\\index{band \
limited functions} function $x_\\\\Omega (t) \\\\in L_2(\\\\mathbb{R})$ is \
one whose Fourier transform $X_\\\\Omega (\\\\omega)$ has support in \
$\\\\left[{-\\\\Omega ,\\\\Omega } \\\\right]$, where $\\\\Omega$ is a \
positive real number. The unitarity of the Fourier transform operation \
ensures that all band limited functions in $L_2(\\\\mathbb{R})$ form a closed \
subspace $L_2^\\\\Omega \\\\left( \\\\mathbb{R} \\\\right)$. The evaluation \
functional (see equations \\\\eqref{Chapter1-17} and \\\\eqref{Chapter1-19}) \
mapping $x_\\\\Omega \\\\to x_\\\\Omega \\\\left( t \\\\right)$ is bounded \
(and therefore continuous), for every function in $L_2^\\\\Omega \\\\left( \
\\\\mathbb{R} \\\\right)$ and all $t$. Aronszajn's theorem (theorem \
\\\\ref{Aronszajn}) then ensures that $L_2^\\\\Omega \\\\left( \\\\mathbb{R} \
\\\\right)$ is a reproducing kernel Hilbert space\\\\index{reproducing \
kernel!Hilbert space}. To find the kernel we use the inverse Fourier \
transform and the band limited property to \
write\\n\\\\begin{equation}\\n\\\\label{Chapter1-82}\\nx_\\\\Omega \\\\left( \
t \\\\right) = {(2 \\\\pi)}^{-1} \\\\int\\\\limits_{ - \\\\Omega }^\\\\Omega \
{X_\\\\Omega \\\\left( \\\\omega \\\\right)} e^{ + i\\\\omega t} d\\\\omega \
~.\\n\\\\end{equation}\\nWe substitute the Fourier transform \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-83}\\nX_\\\\Omega(\\\\omega) = \\\
\\int\\\\limits_{- \\\\infty }^\\\\infty {x_\\\\Omega (t')} e^{-i \\\\omega \
t' } dt' \\n\\\\end{equation}\\nand perform the integral over $\\\\omega$ to \
obtain \\n\\\\begin{equation}\\n\\\\label{Chapter1-84}\\nx_\\\\Omega \
(t)=\\\\int\\\\limits_{-\\\\infty}^\\\\infty {\\\\frac{{\\\\sin \\\\left[ {\\\
\\Omega \\\\left({t-t'} \\\\right)} \\\\right]}}{{\\\\pi \\\\left( {t-t'} \
\\\\right)}}x_\\\\Omega (t') dt'} ~.\\n\\\\end{equation}\\nThis shows the \
reproducing kernel\\\\index{reproducing kernel} property of the associated \
Hilbert space $L_2^\\\\Omega \\\\left( \\\\mathbb{R} \\\\right)$, with a \
real and symmetric kernel function \\\\footnote{Note that the limit \
$\\\\Omega \\\\to \\\\infty$ and using equation \\\\eqref{Chapter1-24} will \
result in the Dirac delta generalized reproducing kernel discussed in section \
\\\\ref{DiracDelta}.}\\n\\\\begin{equation}\\n\\\\label{Chapter1-85}\\nK\\\\\
left( {t,t'} \\\\right) \\\\equiv {{\\\\sin \\\\left[ {\\\\Omega \\\\left( {t \
-t'} \\\\right)} \\\\right]} \\\\mathord{\\\\left/ {\\\\vphantom {{\\\\sin \\\
\\left[ {\\\\Omega \\\\left( {t - t' } \\\\right)} \\\\right]} {\\\\pi \
\\\\left( {t - t'} \\\\right)}}} \\\\right.\\n \
\\\\kern-\\\\nulldelimiterspace} {\\\\pi \\\\left( {t - t'} \\\\right)}} \
~.\\n\\\\end{equation}\\nAnother way to see this result is to note that the \
operator projecting a function into the space $L_2^\\\\Omega \\\\left( \
\\\\mathbb{R} \\\\right)$\\n\\\\begin{equation}\\n\\\\label{Chapter1-86}\\nP_\
\\\\Omega \\\\left\\\\{ {x\\\\left( t \\\\right)} \\\\right\\\\} = {(2 \
\\\\pi)}^{-1} \\\\int\\\\limits_{ - \\\\Omega }^\\\\Omega {X\\\\left( \
\\\\omega \\\\right)e^{i\\\\omega t} d {\\\\omega}} \
\\n\\\\end{equation}\\nis an orthogonal projection. This operator, in the \
frequency domain, has the form \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-87}\\nP_\\\\Omega \\\\left\\\\{ \
{X\\\\left( \\\\omega \\\\right)} \\\\right\\\\} = X\\\\left( \\\\omega \
\\\\right) \\\\times 1_{\\\\left[ { - \\\\Omega ,\\\\Omega } \\\\right]} \
\\\\equiv X_\\\\Omega \\\\left( \\\\omega \
\\\\right)\\n\\\\end{equation}\\nwhere the function $1_{\\\\left[ { - \
\\\\Omega ,\\\\Omega } \\\\right]}$ is defined to be $1$ on the interval \
$\\\\left[ { - \\\\Omega ,\\\\Omega } \\\\right]$ and $0$ elsewhere, and its \
inverse Fourier transform is ${{\\\\sin \\\\left( {\\\\Omega t} \\\\right)} \
\\\\mathord{\\\\left/ {\\\\vphantom {{\\\\sin \\\\left( {\\\\Omega t} \
\\\\right)} {\\\\pi t}}} \\\\right. \\\\kern-\\\\nulldelimiterspace} {\\\\pi \
t}}$. The inverse Fourier transform of $X_\\\\Omega \\\\left( \\\\omega \\\
\\right)$ is the convolution of $x_\\\\Omega (t)$ and ${{\\\\sin \\\\left( \
{\\\\Omega t} \\\\right)} \\\\mathord{\\\\left/ {\\\\vphantom {{\\\\sin \
\\\\left( {\\\\Omega t} \\\\right)} {\\\\pi t}}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {\\\\pi t}}$, which is precisely equation \\\
\\eqref{Chapter1-84} (see equations \\\\eqref{Chapter2-1} and \
\\\\eqref{Chapter2-2} of chapter \\\\ref{Chapter_LTI} for theorems on \
convolution). Thus, the reproducing kernel \\\\eqref{Chapter1-85} is a \
projection operator that maps any function in $L_2 \\\\left( \\\\mathbb{R} \\\
\\right)$ onto the subspace $L_2^\\\\Omega \\\\left( \\\\mathbb{R} \
\\\\right)$. \\n\\nLet us define \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-88}\\np_\\\\tau \\\\left( t \
\\\\right) = {{\\\\sin \\\\left[ {\\\\Omega \\\\left( {t - \\\\tau} \
\\\\right)} \\\\right]} \\\\mathord{\\\\left/ {\\\\vphantom {{\\\\sin \
\\\\left[ {\\\\Omega \\\\left( {t - \\\\tau} \\\\right)} \\\\right]} {\\\\pi \
\\\\left( {t - \\\\tau} \\\\right)}}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {\\\\pi \\\\left( {t - \\\\tau} \\\\right)}} \
= K \\\\left( {t,\\\\tau} \\\\right) ~.\\n\\\\end{equation}\\nWe note that \
$p_\\\\tau \\\\left( t \\\\right) = p_{\\\\tau - t } \\\\left( 0 \\\\right)$ \
and so the functions $p_{\\\\tau}(t)$ are essentially (except for an overall \
multiplicative constant) time shifted versions of a ``mother\\\" function \
${\\\\mathbf{sinc}}\\\\left( {{{\\\\Omega t} \\\\mathord{\\\\left/ \
{\\\\vphantom {{\\\\Omega t} \\\\pi }} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} \\\\pi }} \\\\right)$, \
where\\n\\\\begin{equation}\\n\\\\label{Chapter1-89}\\n{\\\\mathbf{sinc}}\\\\\
left( t \\\\right) \\\\equiv {{\\\\sin \\\\left( {\\\\pi t} \\\\right)} \
\\\\mathord{\\\\left/ {\\\\vphantom {{\\\\sin \\\\left( {\\\\pi t} \
\\\\right)} {\\\\pi t}}} \\\\right.\\n \\\\kern-\\\\nulldelimiterspace} \
{\\\\pi t}} ~.\\n\\\\end{equation}\\nThen we have \\n\\\\begin{equation}\\n\\\
\\label{Chapter1-90}\\n\\\\int\\\\limits_{- \\\\infty }^\\\\infty \
{p_{\\\\tau} \\\\left( {t} \\\\right)p_{{\\\\tau}'} \\\\left( {t} \
\\\\right)dt} = p_{\\\\tau} \\\\left( {{\\\\tau}'} \\\\right) = \
p_{{\\\\tau}'} \\\\left( {\\\\tau} \\\\right) ~.\\n\\\\end{equation}\\nThe \
reproducing kernel equation shows that all elements of $L_2^\\\\Omega \
\\\\left( \\\\mathbb{R} \\\\right)$ can be expanded in terms of the \
continuously parametrized functions $p_\\\\tau(t)$, $\\\\tau \\\\in \
\\\\mathbb{R}$. It is well known that a discretization of the shift \
parameter $\\\\tau$\\n\\\\begin{equation}\\n\\\\label{Chapter1-91} \
\\n\\\\tau_{_{n}} = {{n\\\\pi } \\\\mathord{\\\\left/ {\\\\vphantom \
{{n\\\\pi } \\\\Omega }} \\\\right. \\\\kern-\\\\nulldelimiterspace} \
\\\\Omega } ~,\\n\\\\end{equation}\\nproduces an orthogonal basis \
\\\\cite{Keen88}. The orthogonality follows from setting $\\\\tau = \
{{m\\\\pi} \\\\mathord{\\\\left/ {\\\\vphantom {{m\\\\pi } \\\\Omega }} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} \\\\Omega }$ and ${\\\\tau}' = \
{{n\\\\pi} \\\\mathord{\\\\left/ {\\\\vphantom {{m\\\\pi } \\\\Omega }} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} \\\\Omega }$ in equation \
\\\\eqref{Chapter1-90}. Defining the linear bandwidth variable $B \\\\equiv \
{\\\\Omega \\\\mathord{\\\\left/ {\\\\vphantom {\\\\Omega {2\\\\pi }}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} {2\\\\pi }}$ we are led to the \
orthonormal basis \
functions\\n\\\\begin{equation}\\n\\\\label{Chapter1-92}\\np_n \\\\left( t \\\
\\right)= \\\\sqrt {2B}\\\\;\\\\mbox{sinc}\\\\left[ {2B\\\\left( \
{t-\\\\frac{n}{2B}} \\\\right)} \\\\right] ~.\\n\\\\end{equation}\\nThe \
Fourier transform of $p_n(t)$ is identically zero outside the closed interval \
$\\\\left[ {-2\\\\pi B,+2\\\\pi B} \\\\right]$. Any band limited function $x_\
\\\\Omega\\\\left( t \\\\right)$ can be expressed in terms of this basis\\n\\\
\\begin{equation}\\n\\\\label{Chapter1-93}\\nx_\\\\Omega \\\\left( t \
\\\\right)=\\\\sum\\\\limits_{n=-\\\\infty }^\\\\infty {c_n p_n \\\\left( t \
\\\\right)},\\n\\\\end{equation}\\nwith the expansion coefficients given by \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-94}\\nc_n = \\\\left\\\\langle \
{p_n ,x_\\\\Omega } \\\\right\\\\rangle = x_\\\\Omega \\\\left( {{n \
\\\\mathord{\\\\left/\\n {\\\\vphantom {n {2B}}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {2B}}} \\\\right) \
~.\\n\\\\end{equation}\\nThus we arrive at the sampling \
theorem\\\\index{sampling theorem} \\\\footnote{Although the theorem is \
usually referred to as the Shannon sampling theorem, it was independently \
discovered by Nyquist, Whittaker, and Kotelnikov.} \\\\cite{Shan49, \
Shan98}.\\n\\\\newtheorem{ShannonSamplingTheorem}{Theorem}\\n\\\\begin{thm}\\\
n\\\\label{SamplingTheorem}\\nLet $x\\\\left( t \\\\right)$ be a continuous \
time signal whose Fourier transform $X\\\\left( \\\\omega \\\\right)$ is zero \
outside the interval $\\\\left[ {-2\\\\pi B,+2\\\\pi B} \\\\right]$, i.e., \
whose linear frequency content is limited to the band $\\\\left[ {-B,+B} \
\\\\right]\\\\ $. If $\\\\Delta T$ is a positive constant satisfying the \
following inequalities \\n\\\\begin{equation}\\n\\\\label{Chapter1-95-0}\\n0 \
< 2B ~ \\\\Delta T \\\\le 1~,\\n\\\\end{equation}\\nthen $x(t)$ can be \
uniquely reconstructed from its sampled values $x(n \\\\Delta T)$, $n \\\\in \
\\\\mathbb{Z}$ by the following interpolation formula \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-95}\\nx\\\\left( t \\\\right) = \
2B ~ \\\\Delta T\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {x\\\\left( \
{n\\\\Delta T} \\\\right)\\\\frac{{\\\\sin \\\\left[ {2\\\\pi B\\\\left( {t - \
n\\\\Delta T} \\\\right)} \\\\right]}}{{2\\\\pi B\\\\left( {t - n\\\\Delta T} \
\\\\right)}}} ~.\\n\\\\end{equation}\\nCondition \\\\eqref{Chapter1-95-0} is \
usually written in terms of the sampling frequency $f_s \\\\equiv {1 \
\\\\mathord{\\\\left/ {\\\\vphantom {1 {\\\\Delta T}}} \\\\right. \\\\kern-\\\
\\nulldelimiterspace} {\\\\Delta T}}$ in the form\\n\\\\[\\nf_s \\\\ge 2B ~,\
\\n\\\\]\\nand stated as the requirement that the signal sampling frequency \
should equal to or exceed twice the highest frequency of the \
signal.\\n\\\\end{thm}\\nThe sampling theorem \\\\ref{SamplingTheorem} \
applied to the $\\\\mathbf{sinc}$ function itself, at a sampling frequency \
equal to twice the highest frequency, yields the following \
result\\n\\\\begin{equation}\\n\\\\label{Chapter1-96}\\n\\\\frac{{\\\\sin \
\\\\left[ {2\\\\pi B\\\\left( {t - t'} \\\\right)} \\\\right]}}{{2\\\\pi \
B\\\\left( {t - t'} \\\\right)}} = \\\\sum\\\\limits_{n = - \\\\infty \
}^\\\\infty {\\\\frac{{\\\\sin \\\\left[ {2\\\\pi B\\\\left( {t - \
\\\\frac{n}{{2B}}} \\\\right)} \\\\right]}}{{2\\\\pi B\\\\left( {t - \
\\\\frac{n}{{2B}}} \\\\right)}}\\\\frac{{\\\\sin \\\\left[ {2\\\\pi \
B\\\\left( {t' - \\\\frac{n}{{2B}}} \\\\right)} \\\\right]}}{{2\\\\pi \
B\\\\left( {t' - \\\\frac{n}{{2B}}} \
\\\\right)}}},\\n\\\\end{equation}\\nwhich shows the reproducing kernel as an \
infinite sum over the products of the orthogonal basis functions. This is an \
example of theorem \\\\ref{BergmanKernel} and equation \
\\\\eqref{Chapter1-41}.\\n\\n\\n\\\\section[The Basis Operator]{The Basis \
Operator in $L_2 ( \\\\mathbb{R} )$}\\n\\nThe correspondence between a \
function and its transform coefficients with respect to an orthonormal basis \
expressed by equations \\\\eqref{Chapter1-39} and \\\\eqref{Chapter1-40} can \
be interpreted in terms of linear operators \\\\cite{DuSc52, DaGM86, Daub92}. \
Consider the linear operator \\n\\\\begin{equation}\\n\\\\label{Chapter1-97}\
\\nT_\\\\phi :L_2 \\\\left( \\\\mathbb{R} \\\\right) \\\\to l_2 \\\\left( \\\
\\mathbb{Z} \\\\right) ~,\\n\\\\end{equation}\\nassociated with a basis (not \
necessarily orthonormal) $\\\\left\\\\{ {\\\\phi _n \\\\left( t \\\\right),n \
\\\\in \\\\mathbb{Z}} \\\\right\\\\}$, defined by \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-98}\\nT_\\\\phi \\\\left\\\\{ {x\
\\\\left( t \\\\right)} \\\\right\\\\} = \\\\left[ { \\\\ldots \
,\\\\left\\\\langle {{\\\\phi _n }}\\n \\\\mathrel{\\\\left | {\\\\vphantom \
{{\\\\phi _n } x}} \\\\right. \\\\kern-\\\\nulldelimiterspace}\\n {x} \
\\\\right\\\\rangle , \\\\ldots } \\\\right]^T ~.\\n\\\\end{equation}\\nIt \
is common to refer to this expansion as the analysis formula, or the forward \
transform, and to the linear operator as the analysis operator of the \
associated basis. The corresponding adjoint operator, the synthesis \
operator, is a linear map \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-99}\\n{T^+}_\\\\phi :l_2 \
\\\\left( \\\\mathbb{Z} \\\\right) \\\\to L_2 \\\\left( \\\\mathbb{R} \
\\\\right) ~,\\n\\\\end{equation}\\ndefined by the synthesis formula, or the \
inverse transform, \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-100}\\n{T^+}_\\\\phi ~ \\\\left\\\
\\{ {\\\\underline{c}} \\\\right\\\\} = \\\\sum\\\\limits_{n = - \\\\infty \
}^\\\\infty {c_n \\\\phi _n \\\\left( t \\\\right)} ,~~ \\\\underline{c} \
\\\\equiv \\\\left[ { \\\\ldots , c_n , \\\\ldots } \\\\right]^T \
~.\\n\\\\end{equation}\\nThe basis operator\\\\index{operator!basis} is then \
defined by\\n\\\\begin{equation}\\n\\\\label{Chapter1-101}\\n{T^+}_\\\\phi T_\
\\\\phi :L_2 \\\\left( \\\\mathbb{R} \\\\right) \\\\to L_2 \\\\left( \
\\\\mathbb{R} \\\\right)\\n\\\\end{equation}\\nand can be expressed in the \
Dirac notation together with the summation convention, \
as\\n\\\\begin{equation}\\n\\\\label{Chapter1-102}\\n{T^+}_\\\\phi T_\\\\phi \
= {\\\\left| {\\\\phi _n } \\\\right\\\\rangle } \\\\left\\\\langle {\\\\phi \
_n } \\\\right| ~, \\n\\\\end{equation}\\nor can be defined by its action on \
any function $x(t)$ (note the summation convention in the Dirac \
notation)\\n\\\\begin{equation}\\n\\\\label{Chapter1-103}\\n{T^+}_\\\\phi \
T_\\\\phi ~ \\\\left\\\\{ {x\\\\left( t \\\\right)} \\\\right\\\\} = \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left\\\\langle {\\\\phi \
_n ,x} \\\\right\\\\rangle \\\\phi _n \\\\left( t \\\\right)} = \\\\left| {\\\
\\phi _n } \\\\right\\\\rangle \\\\left\\\\langle {{\\\\phi _n }} \
\\\\mathrel{\\\\left | {\\\\vphantom {{\\\\phi _n } x}}\\n \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle \
~.\\n\\\\end{equation}\\nThe basis operator ${T^+}_\\\\phi T_\\\\phi$ is \
clearly self adjoint and positive definite, and so we \
have\\n\\\\begin{equation}\\n\\\\label{Chapter1-104}\\n\\\\left\\\\| \
{{T^+}_\\\\phi T_\\\\phi } \\\\right\\\\|_2 = \\\\rho _{{T^+}_\\\\phi T_\\\
\\phi} = \\\\lambda _{\\\\max } \
\\\\end{equation}\\nand\\n\\\\begin{equation}\\n\\\\label{Chapter1-105}\\n\\\\\
left\\\\| {\\\\left( {{T^+}_\\\\phi T_\\\\phi } \\\\right)^{ - 1} } \
\\\\right\\\\|_2 = \\\\lambda _{\\\\min }^{ - 1} ~ \
.\\n\\\\end{equation}\\nwhere $\\\\lambda_{max}$ and $\\\\lambda_{min}$ are \
the largest and smallest eigenvalues of the basis operator $T_\\\\phi ^ + T_\
\\\\phi $. The ratio ${{\\\\lambda _{{\\\\rm max}} } \\\\mathord{\\\\left/ {\
\\\\vphantom {{\\\\lambda _{{\\\\rm max}} } {\\\\lambda _{{\\\\rm min}} }}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} {\\\\lambda _{{\\\\rm min}} }}$ \
is the condition number\\\\index{condition number} of the same operator and \
its size is crucial in calculating the inverse operator in a numerically \
stable manner. \\n\\nIf the basis functions are not orthonormal, as is the \
case in an over complete set in which the elements are not linearly \
independent, the numerical inversion of the basis operator ${T^+}_\\\\phi \
T_\\\\phi $ depends on how large its condition number is. If, on the other \
hand, the analysis operator $T_\\\\phi $ is unitary then ${T^+}_\\\\phi T_\\\
\\phi = \\\\mathbf{1}$, all the eigenvalues of the basis operator are equal \
to $1$ and the inverse is guaranteed to exist. A unitary analysis operator \
indicates an orthonormal basis and for such an operator we \
have\\n\\\\begin{equation}\\n\\\\label{Chapter1-106}\\n\\\\left\\\\| x \
\\\\right\\\\|^2 = \\\\left\\\\langle {x,x} \\\\right\\\\rangle = \
\\\\left\\\\langle {x,{T^+}_\\\\phi T_\\\\phi x} \\\\right\\\\rangle = \
\\\\left\\\\langle {T_\\\\phi x,T_\\\\phi x} \\\\right\\\\rangle = \\\\sum\
\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left| {\\\\left\\\\langle \
{\\\\phi _n ,x} \\\\right\\\\rangle } \\\\right|^2 \
},\\n\\\\end{equation}\\nwhich is Parseval's relation\\\\index{Parseval's \
relation} \\\\eqref{Chapter1-37} \\\\footnote{In the Dirac notation $\\\\left\
\\\\| x \\\\right\\\\|^2 = \\\\left\\\\langle x \\\\right|T_\\\\phi ^ + \
T_\\\\phi \\\\left| x \\\\right\\\\rangle = \\\\left\\\\langle {x} \
\\\\mathrel{\\\\left | {\\\\vphantom {x {\\\\phi _n }}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace}\\n {{\\\\phi _n }} \\\\right\\\\rangle \
\\\\left\\\\langle {{\\\\phi _n }} \\\\mathrel{\\\\left | {\\\\vphantom \
{{\\\\phi _n } x}} \\\\right. \\\\kern-\\\\nulldelimiterspace} {x} \
\\\\right\\\\rangle \\\\equiv \\\\sum\\\\limits_{n = - \\\\infty \
}^\\\\infty {\\\\left| {\\\\left\\\\langle {{\\\\phi _n }} \
\\\\mathrel{\\\\left | {\\\\vphantom {{\\\\phi _n } x}}\\n \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle } \\\\right|^2 }$.}.\
\\n\\n\\n\\\\section[Biorthogonal Bases]{Biorthogonal Bases and \
Representations in $L_2 \\\\left( \\\\mathbb{R} \\\\right)$} \
\\\\label{SectionBiorthogonal}\\n\\n\\nMore general than an orthogonal basis \
is the concept of biorthogonal bases\\\\index{basis!biorthogonal} that occur \
in the definition of a Riesz basis. As we shall see later, biorthogonal \
bases allow more freedom in choosing wavelet functions of compact support. \
The idea of biorthogonality is best explained for finite dimensional vector \
spaces. Consider the xy-plane and the a pair of vectors $\\\\mathbf{v}_1 \\\
\\equiv \\\\left[ {0,1} \\\\right]^T$ and $\\\\mathbf{v}_2 \\\\equiv \
\\\\left[ {1 ,\\\\tan \\\\theta } \\\\right]^T$, and a second pair \
$\\\\mathbf{w}_1 \\\\equiv \\\\left[ {- \\\\tan \\\\theta ,1 } \\\\right]^T$ \
and $\\\\mathbf{w}_2 \\\\equiv \\\\left[ {1,0} \\\\right]^T$ for some angle \
$0 < \\\\theta < {\\\\pi \\\\mathord{\\\\left/ {\\\\vphantom {\\\\pi 2}} \
\\\\right. \\\\kern-\\\\nulldelimiterspace} 2}$. Then $\\\\left\\\\langle \
{\\\\mathbf{w}_i , \\\\mathbf{v}_j } \\\\right\\\\rangle \\\\equiv \
\\\\mathbf{w}_i^T \\\\mathbf{v}_j = \\\\delta_{ij}$ and we call the two \
pairs biorthogonal. The two pair of vectors are shown in figure \
\\\\ref{fig:fig-Chapter1-3} for $\\\\theta = 60^ \\\\circ$. \
\\n\\\\begin{figure}[h!]\\n\\t\\\\begin{centering}\\n\\t\\\\includegraphics[\
width=2in]{./fig-Chapter1-3.pdf}\\n\\t\\\\caption{Two biorthogonal pairs of \
vectors.}\\n\\t\\\\label{fig:fig-Chapter1-3}\\n\\t\\\\end{centering}\\n\\\\\
end{figure}\\nNow any vector $\\\\mathbf{x} = \\\\left[ {x,y} \\\\right]^T$ \
can be expanded either in terms of the first pair, or in terms of the second \
pair. The coefficients of expansion are then given by the inner product of \
the given vector with the alternate pair. For instance, if $\\\\mathbf{x} = \
\\\\alpha_1 \\\\mathbf{v}_1 + \\\\alpha_2 \\\\mathbf{v}_2$, then \
$\\\\alpha_j = \\\\left\\\\langle {\\\\mathbf{w}_j , \\\\mathbf{x}} \\\\right\
\\\\rangle$. Alternatively, if $\\\\mathbf{x} = \\\\beta_1 \\\\mathbf{w}_1 \
+ \\\\beta_2 \\\\mathbf{w}_2$, then $\\\\beta_j = \\\\left\\\\langle \
{\\\\mathbf{v}_j , \\\\mathbf{x}} \\\\right\\\\rangle$ for $j = 1,2$. \
Neither pair $\\\\left\\\\{ {\\\\mathbf{v}_1 , \\\\mathbf{v}_2 } \
\\\\right\\\\}$, nor $\\\\left\\\\{ {\\\\mathbf{w}_1 , \\\\mathbf{w}_2 } \
\\\\right\\\\}$, however, is an orthogonal \
set.\\n\\n\\\\newtheorem{RieszBais}{Definition}\\n\\\\begin{defn}\\n\\\\label{\
RieszBais}\\nGiven a complete and orthonormal basis $\\\\left\\\\{ \\\\phi _n \
\\\\left( t \\\\right), ~ n \\\\in \\\\mathbb{Z} \\\\right\\\\}$, a new set \
of basis functions, a Riesz basis\\\\index{basis!Riesz}, is defined through a \
non unitary but invertible continuous linear operator $L$ (also known as a \
topological isomorphism) by the relations \
\\\\cite{Chri03}\\n\\\\begin{equation}\\n\\\\label{Chapter1-107}\\n\\\\xi _n \
\\\\left( t \\\\right) = L\\\\phi _n \\\\left( t \\\\right) \
\\\\leftrightarrow \\\\left| {\\\\xi _n } \\\\right\\\\rangle = L\\\\left| {\
\\\\phi _n } \\\\right\\\\rangle \
~.\\n\\\\end{equation}\\n\\\\end{defn}\\n\\\\noindent Since $L$ is not \
unitary, the new basis functions are not orthonormal \\\\footnote{If $L$ were \
unitary, i.e., $L^+ L =1$, then $\\\\left\\\\langle {\\\\phi _m ,\\\\phi _n } \
\\\\right\\\\rangle = \\\\left\\\\langle {L^ + L\\\\phi _m ,\\\\phi _n } \\\
\\right\\\\rangle = \\\\left\\\\langle {\\\\xi _m ,\\\\xi _n } \
\\\\right\\\\rangle = \\\\delta_{mn}.$}. We have \\n\\\\begin{equation}\\n\\\
\\label{Chapter1-108}\\n\\\\left\\\\langle {\\\\phi _m ,\\\\phi _n } \
\\\\right\\\\rangle = \\\\delta_{mn} = \\\\left\\\\langle {L^{ - 1} \
L\\\\phi _m ,\\\\phi_n } \\\\right\\\\rangle = \\\\left\\\\langle {L\\\\phi \
_m ,\\\\left( {L^{ - 1} } \\\\right)^ + \\\\phi_n } \\\\right\\\\rangle \
\\n\\\\end{equation}\\nSince $L$ is a topological isomorphism, its inverse is \
also a continuous linear map and it can be used to define a second Riesz \
basis set $\\\\left\\\\{ \\\\chi_n \\\\left( t \\\\right), ~ n \\\\in \
\\\\mathbb{Z} \\\\right\\\\}$\\n\\\\begin{equation}\\n\\\\label{Chapter1-109}\
\\n\\\\chi _n \\\\left( t \\\\right) \\\\equiv \\\\left( {L^{ - 1} } \
\\\\right)^ + \\\\phi_n \\\\left( t \\\\right) \\\\leftrightarrow \\\\left| \
{\\\\chi _n } \\\\right\\\\rangle = \\\\left( {L^{ - 1} } \\\\right)^ + \
\\\\left| {\\\\phi _n } \\\\right\\\\rangle ~.\\n\\\\end{equation}\\nIt is \
easy to see that \\n\\\\begin{equation}\\n\\\\label{Chapter1-110}\\n\\\\left\
\\\\langle {{\\\\xi_m }} \\\\mathrel{\\\\left | {\\\\vphantom {{\\\\xi _m } {\
\\\\chi_n }}} \\\\right. \\\\kern-\\\\nulldelimiterspace} {{\\\\chi_n }} \
\\\\right\\\\rangle = \\\\delta_{mn},\\n\\\\end{equation}\\nwhich shows the \
biorthogonal property of the two Riesz bases $\\\\left\\\\{ {\\\\xi _n \
\\\\left( t \\\\right)} \\\\right\\\\}$ and $\\\\left\\\\{ {\\\\chi _n \
\\\\left( t \\\\right)} \\\\right\\\\}$. The corresponding analysis \
operators are denoted by $T_\\\\xi $ and $T_\\\\chi $, \
where\\n\\\\begin{equation}\\n\\\\label{Chapter1-111}\\nT_\\\\xi x\\\\left( \
t \\\\right) = \\\\left[ { \\\\ldots ,\\\\left\\\\langle {\\\\xi _n ,x} \
\\\\right\\\\rangle , \\\\ldots } \\\\right]^T ,~T_\\\\chi x\\\\left( t \
\\\\right) = \\\\left[ { \\\\ldots ,\\\\left\\\\langle {\\\\chi _n ,x} \
\\\\right\\\\rangle , \\\\ldots } \\\\right]^T .\\n\\\\end{equation}\\nAny \
element of $L_2(\\\\mathbb{R})$ can now be represented in two equivalent ways \
in terms of the two biorthogonal sets. Using the Dirac notation and the \
summation convention we have the following expansion in terms of the original \
orthonormal set\\n\\\\[\\n\\\\left| x \\\\right\\\\rangle = \\\\left| \
{\\\\phi_n } \\\\right\\\\rangle \\\\left\\\\langle {{\\\\phi_n}} \
\\\\mathrel{\\\\left | {\\\\vphantom {{\\\\phi_n } x}} \\\\right. \\\\kern-\\\
\\nulldelimiterspace} {x} \\\\right\\\\rangle ~,\\n\\\\]\\nwhich upon acting \
on by the operator $L$ and using equation \\\\eqref{Chapter1-107} \
becomes\\n\\\\[\\nL\\\\left| x \\\\right\\\\rangle = L\\\\left| {\\\\phi _n \
} \\\\right\\\\rangle \\\\left\\\\langle {{\\\\phi _n }} \\\\mathrel{\\\\left \
| {\\\\vphantom {{\\\\phi _n } x}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle = \\\\left| {\\\\xi \
_n } \\\\right\\\\rangle \\\\left\\\\langle {{\\\\phi _n }} \
\\\\mathrel{\\\\left | {\\\\vphantom {{\\\\phi _n } x}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} {x} \\\\right\\\\rangle ~.\\n\\\\]\\nNow \
using the adjoint of equation \\\\eqref{Chapter1-109} we write \
\\n\\\\[\\n\\\\left\\\\langle {\\\\phi_n } \\\\right| = \\\\left\\\\langle \
{\\\\chi_n } \\\\right|L ~,\\n\\\\]\\nand so for any $x(t)$ we have \
\\n\\\\[\\nL\\\\left| x \\\\right\\\\rangle = \\\\left| {\\\\xi _n } \
\\\\right\\\\rangle \\\\left\\\\langle {\\\\chi _n } \\\\right|L\\\\left| x \
\\\\right\\\\rangle ~,\\n\\\\]\\nfrom which we arrive at the completeness \
relation\\n\\\\begin{equation}\\n\\\\label{Chapter1-112}\\n{\\\\left| {\\\\xi \
_n } \\\\right\\\\rangle \\\\left\\\\langle {\\\\chi _n } \\\\right|} = \
\\\\mathbf{1} ~,\\n\\\\end{equation}\\nwith the associated expansion \
formula\\n\\\\begin{equation}\\n\\\\label{Chapter1-113}\\n\\\\left| x \
\\\\right\\\\rangle = \\\\left\\\\langle {{\\\\xi_n }} \\\\mathrel{\\\\left \
| {\\\\vphantom {{\\\\xi_n } x}} \\\\right. \\\\kern-\\\\nulldelimiterspace} \
{x} \\\\right\\\\rangle \\\\left| {\\\\chi_n } \\\\right\\\\rangle \
.\\n\\\\end{equation}\\nOn the other hand, starting with the identity \
\\n\\\\[\\nL^+ \\\\left| x \\\\right\\\\rangle = \\\\left| {\\\\phi_n } \
\\\\right\\\\rangle \\\\left\\\\langle {\\\\phi_n } \\\\right|L^+ \\\\left| \
x \\\\right\\\\rangle ~,\\n\\\\]\\nwe arrive at the completeness \
relation\\n\\\\begin{equation}\\n\\\\label{Chapter1-114}\\n{\\\\left| \
{\\\\chi _n } \\\\right\\\\rangle \\\\left\\\\langle {\\\\xi _n } \\\\right|} \
= \\\\mathbf{1} ~,\\n\\\\end{equation}\\nwith the associated expansion \
formula\\n\\\\begin{equation}\\n\\\\label{Chapter1-115}\\n\\\\left| x \
\\\\right\\\\rangle = \\\\left\\\\langle {{\\\\chi_n }} \\\\mathrel{\\\\left \
| {\\\\vphantom {{\\\\chi_n } x}} \\\\right. \\\\kern-\\\\nulldelimiterspace} \
{x} \\\\right\\\\rangle \\\\left| {\\\\xi_n } \\\\right\\\\rangle \
.\\n\\\\end{equation}\\n\\nThus, biorthogonal Riesz bases are more general \
than orthogonal bases. We will use biorthogonal bases in the construction of \
biorthogonal wavelets\\\\index{biorthogonal!wavelets} of compact support \
whose implementation does not require filter sequences of even length; both \
even and odd lengths will be permitted in this context. Even more general is \
the concept of a frame that will be introduced in the next two sections. It \
is important to note, however, that both orthogonal and Riesz bases have the \
property that removing a single element from either set will destroy their \
completeness; such bases are known as exact\\\\index{basis!exact}. Riesz \
bases are also exact frames \\\\cite{Youn80}. Before we introduce the \
concept of a frame in an infinite dimensional Hilbert space we will, in the \
next section, look at the problem of vector representation in a finite \
dimensional vector space, e.g., $\\\\mathbb{C}^n$. \
\\n\\n\\n\\n\\\\section{Frames\\\\index{frame} in A Finite Dimensional Vector \
Space}\\n\\nA vector $\\\\mathbf{x} \\\\in \\\\mathbb{C}^n$ is represented by \
the column $\\\\left[ {x_1 , \\\\ldots ,x_n } \\\\right]^T $ where $x_k \
\\\\in \\\\mathbb{C}$, $k = 1, \\\\ldots ,n$. The inner product is defined \
by\\n\\\\[\\n\\\\left\\\\langle {\\\\mathbf{z},\\\\mathbf{x}} \
\\\\right\\\\rangle \\\\equiv \\\\mathbf{z}^ + \\\\mathbf{x} = \
\\\\sum\\\\limits_{k = 1}^n {\\\\mathbf{z}^*_{k} \\\\mathbf{x}_{_k}} , ~~ \\\
\\mathbf{z}, \\\\mathbf{x} \\\\in \\\\mathbb{C}^n~.\\n\\\\]\\nConsider a \
linear mapping (operator) between $\\\\mathbb{C}^n$ and $\\\\mathbb{C}^m$, $n \
\\\\ge m$, represented by a complex $m \\\\times n$ matrix $\\\\mathbf{A}$, \
i.e., $\\\\mathbf{A} \\\\mathbf{x} = \\\\mathbf{y}$, $\\\\mathbf{x} \\\\in \\\
\\mathbb{C}^n$, $\\\\mathbf{y} \\\\in \\\\mathbb{C}^m$. Denoting the columns \
of $\\\\mathbf{A}$ by $\\\\mathbf{c}_k$, $k = 1, \\\\ldots ,n$, it is easy to \
see that\\n\\\\begin{equation}\\n\\\\label{Chapter1-116}\\n\\\\mathbf{A} \
\\\\mathbf{x} = \\\\sum\\\\limits_{k = 1}^n {x_{k} \\\\mathbf{c}_{k}}~, \\n\\\
\\end{equation}\\nthat is, the range space of the matrix $\\\\mathbf{A}$, \
denoted by $\\\\mathscr{R}(\\\\mathbf{A}) $ is the space spanned by the \
columns of the same matrix. If we assume that \
$\\\\mathscr{R}(\\\\mathbf{A})$ is all of $\\\\mathbb{C}^m$, $m \\\\le n$, \
then there are $m$ independent columns of $\\\\mathbf{A}$. The column rank\\\
\\index{column rank} of the matrix $\\\\mathbf{A}$ is the number of \
independent columns of the matrix, and defines the dimension of the range \
space $\\\\mathscr{R}(\\\\mathbf{A})$. \\n\\nThe adjoint map (operator) \
denoted by $\\\\mathbf{A}^ +$ is defined between $\\\\mathbb{C}^m$ and \
$\\\\mathbb{C}^n$ and is the $n \\\\times m$ matrix $\\\\left( \
{\\\\mathbf{A}^ * }\\\\right)^T $ (the Hermitian conjugate of \
$\\\\mathbf{A}$). The dimension of the range space $\\\\mathscr{R} \
(\\\\mathbf{A}^ +)$ is the column rank of the matrix $\\\\mathbf{A}^ +$ which \
is equal to the row rank\\\\index{row rank} of the matrix $\\\\mathbf{A}$ \
\\\\footnote{To be precise, the row rank of $\\\\mathbf{A}^*$, which is the \
same as the row rank of $\\\\mathbf{A}$.}, which is also equal to the column \
rank of the same matrix. The rank $r_{\\\\mathbf{A}}$ of the matrix \
$\\\\mathbf{A}$ is the number of linearly independent columns (or rows) of \
that matrix. The following theorem specifies the dimensions of the various \
range and null spaces \\\\cite{Stra88}.\\n\\\\newtheorem{MatrixRank}{Theorem}\
\\n\\\\begin{thm}\\nIf a complex $m \\\\times n$ matrix $\\\\mathbf{A}$ has \
rank $r_{\\\\mathbf{A}}$ then,\\n\\\\begin{itemize}\\n\\t\\\\item \
$\\\\mathscr{R} (\\\\mathbf{A})$ and $\\\\mathscr{R} (\\\\mathbf{A}^+)$ have \
dimension $r_{\\\\mathbf{A}}$,\\n\\t\\\\item $\\\\mathscr{N} \
(\\\\mathbf{A})$ has dimension $(n-r_{\\\\mathbf{A}})$ while $\\\\mathscr{N} \
(\\\\mathbf{A}^+)$ has dimension \
$(m-r_{\\\\mathbf{A}})$.\\n\\\\end{itemize}\\n\\\\end{thm}\\n\\nWhen $m=n=r_{\
\\\\mathbf{A}}$, the matrix $\\\\mathbf{A}$ is square and has full rank. It \
is, therefore, non singular (has non zero determinant) and invertible. \
Consequently, an equation of the \
form\\n\\\\begin{equation}\\n\\\\label{Chapter1-117}\\n\\\\mathbf{A} \
\\\\mathbf{x} = \\\\mathbf{y}, ~ \\\\mathbf{x} \\\\in \\\\mathbb{C}^m,~ \
\\\\mathbf{y} \\\\in \\\\mathbb{C}^n, \\n\\\\end{equation}\\nhas the unique \
solution $\\\\mathbf{x} = \\\\mathbf{A}^{-1} \\\\mathbf{y}$ when \
$m=n=r_{\\\\mathbf{A}}$. More generally, a solution exists only if \
$\\\\mathbf{y} \\\\in \\\\mathscr{R} (\\\\mathbf{A})$, i.e., if \
$\\\\mathbf{y}$ is a linear combination of the columns of $\\\\mathbf{A}$. \
By theorem \\\\ref{AandAPlusSpaces} we have $\\\\mathscr{R} (\\\\mathbf{A}) = \
(\\\\mathscr{N} ({\\\\mathbf{A}}^+))^ \\\\bot$ and so if $\\\\mathbf{y} \
\\\\in \\\\mathscr{R} (\\\\mathbf{A})$ then $\\\\mathbf{y}$ must be \
orthogonal to $\\\\mathscr{N} ({\\\\mathbf{A}}^+)$. In other words, the \
matrix equation \\\\eqref{Chapter1-117} has a solution if, and only if, \
$\\\\left\\\\langle {\\\\mathbf{z}, \\\\mathbf{y}} \\\\right\\\\rangle = 0$ \
for every $\\\\mathbf{z} \\\\in {\\\\mathbb{C}}^n$ that satisfies \
${\\\\mathbf{A}}^+ \\\\mathbf{z} = \\\\mathbf{0}$. \\n\\nIf a solution to \\\
\\eqref{Chapter1-117} exists then it is unique if, and only if, \
$\\\\mathscr{N}(\\\\mathbf{A}) = \\\\left\\\\{ \\\\mathbf{0} \\\\right\\\\}$. \
Thus, when $m \\\\le n$ and $r_{\\\\mathbf{A}} < Min\\\\left( {m,n} \
\\\\right)$ the dimension of the null space $\\\\mathscr{N} (\\\\mathbf{A})$ \
is greater than zero and so if a solution to \\\\eqref{Chapter1-117} exists, \
it is not unique. When multiple solutions exist, a unique solution whose \
norm is a minimum can be found using the following theorem \\\\cite{CaMe79, \
GoVL96}.\\n\\\\newtheorem{Pseudoinverse}{Theorem}\\n\\\\begin{thm}\\nLet \
$\\\\mathbf{A}$ be a complex $m \\\\times n$ matrix, $m \\\\le n$. Then the \
unique solution to the equation $\\\\mathbf{A} \\\\mathbf{x} = \\\\mathbf{y}$ \
whose norm $\\\\left\\\\| x \\\\right\\\\|$ is the smallest among all the \
solutions is $\\\\hat {\\\\mathbf{x}} = {\\\\mathbf{A}^+} \\\\mathbf{u}$ \
where $\\\\mathbf{u}$ is any solution to the equation \
$\\\\mathbf{A}\\\\mathbf{A}^+ \\\\mathbf{u}=\\\\mathbf{y}$. In particular, \
when $r_{\\\\mathbf{A}}=m \\\\le n$, the minimum norm solution \
is\\n\\\\begin{equation}\\n\\\\label{Chapter1-118}\\n\\\\hat {\\\\mathbf{x}} \
= {\\\\mathbf{A}^+} \\\\left( {\\\\mathbf{A} {\\\\mathbf{A}^ +}} \
\\\\right)^{-1} \\\\mathbf{y} \\\\equiv {\\\\mathbf{A}_R ^\\\\dag} \
\\\\mathbf{y},\\n\\\\end{equation}\\nwhich also defines the right pseudo \
inverse\\\\index{pseudo inverse} $\\\\mathbf{A}_R ^\\\\dag$ \
satisfying\\n\\\\begin{equation}\\n\\\\label{Chapter1-119}\\n\\\\mathbf{A} \\\
\\mathbf{A}_R^\\\\dag = \\\\mathbf{1}.\\n\\\\end{equation}\\nFurthermore, \
if $\\\\mathbf{B}$ is a complex $n \\\\times m$ matrix, $n \\\\ge m$, whose \
rank is $m$, then the equation \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-120}\\n\\\\mathbf{B} \
\\\\mathbf{y} = \\\\mathbf{x}, ~ \\\\mathbf{y} \\\\in \\\\mathbb{C}^m,~ \
\\\\mathbf{x} \\\\in \\\\mathbb{C}^n, \\n\\\\end{equation}\\nhas no solution. \
However, there is a unique vector $\\\\hat{\\\\mathbf{y}}$ that minimizes \
the norm $\\\\left\\\\| {\\\\mathbf{B} \\\\mathbf{y} - \\\\mathbf{x}} \
\\\\right\\\\|$ (for $l_2$ norm this is known as the least squares solution), \
namely, \\n\\\\begin{equation}\\n\\\\label{Chapter1-121}\\n\\\\hat \
{\\\\mathbf{y}} = \\\\left( {\\\\mathbf{B}^ + \\\\mathbf{B}} \\\\right)^{-1} \
\\\\mathbf{B}^ + \\\\mathbf{x} \\\\equiv \\\\mathbf{B}_L^\\\\dag \
\\\\mathbf{x}\\n\\\\end{equation}\\nwhich also defines the left pseudo \
inverse $\\\\mathbf{B}_L ^\\\\dag$ \
satisfying\\n\\\\begin{equation}\\n\\\\label{Chapter1-122}\\n\\\\mathbf{B}_L^\
\\\\dag \\\\mathbf{B} = \\\\mathbf{1}.\\n\\\\end{equation}\\n\\\\end{thm}\\n\
\\n\\\\noindent Now consider the \
identity\\n\\\\begin{equation}\\n\\\\label{Chapter1-123}\\n\\\\mathbf{y} = \\\
\\mathbf{A} \\\\mathbf{A}^+ \\\\left( {\\\\mathbf{A} \\\\mathbf{A}^+} \
\\\\right)^{-1} \\\\mathbf{y} = \\\\mathbf{A} \\\\mathbf{A}_R^\\\\dag \
\\\\mathbf{y}, ~~~ \\\\mathbf{y} \\\\in \\\\mathbb{C}^m \
~,\\n\\\\end{equation}\\nor \
equivalently\\n\\\\begin{equation}\\n\\\\label{Chapter1-124}\\ny_{_k} = \
\\\\sum\\\\limits_{j = 1}^n {\\\\sum\\\\limits_{l = 1}^m {\\\\mathbf{A}_{kj} \
\\\\left[ {\\\\mathbf{A}_R^ \\\\dag } \\\\right]_{jl} y_{_l} } }, ~~k = 1, \\\
\\ldots ,m.\\n\\\\end{equation}\\nWe write\\n\\\\[\\n\\\\sum\\\\limits_{l = \
1}^m {\\\\left[ {\\\\mathbf{A}_R^ \\\\dag} \\\\right]_{jl} y_{_l} } = \
\\\\sum\\\\limits_{l = 1}^m {\\\\left[ {\\\\left( {\\\\mathbf{c}_j^ \\\\dag} \
\\\\right)^ * } \\\\right]_{l} y_{_l} } \\\\equiv \\\\left\\\\langle \
{\\\\mathbf{c}_j^ \\\\dag ,\\\\mathbf{y}} \
\\\\right\\\\rangle,\\n\\\\]\\nwhere\\n\\\\[\\n\\\\left[ \
{\\\\left({\\\\mathbf{c}_j^ \\\\dag} \\\\right)} \\\\right]_l = \\\\left[ \
{\\\\left( {\\\\mathbf{A}_R^ \\\\dag} \\\\right)^ *} \\\\right]_{jl} = \
\\\\left[ {\\\\left( {\\\\mathbf{A}_R^ \\\\dag} \\\\right)^ +} \
\\\\right]_{lj}.\\n\\\\]\\nThus, $\\\\mathbf{c}_j^ \\\\dag$ is the $j$-th \
column of the $n \\\\times m$ matrix ${\\\\left( {\\\\mathbf{A}_R^ \\\\dag} \
\\\\right)^ +}$,\\n\\\\[\\n\\\\left( {\\\\mathbf{A}_R^ \\\\dag} \\\\right)^ + \
\\\\equiv \\\\left[ {\\\\mathbf{c}_1^ \\\\dag , \\\\ldots , \
\\\\mathbf{c}_n^ \\\\dag} \\\\right],\\n\\\\]\\nand equation \
\\\\eqref{Chapter1-124} can be written as\\n\\\\[\\ny_{_k} = \
\\\\sum\\\\limits_{j = 1}^n {\\\\mathbf{A}_{kj} \\\\left\\\\langle \
{\\\\mathbf{c}_j^ \\\\dag, \\\\mathbf{y}} \\\\right\\\\rangle }, ~ k = 1, \
\\\\ldots ,m,\\n\\\\]\\nwhich in view of the relation \\\\eqref{Chapter1-116} \
becomes\\n\\\\begin{equation}\\n\\\\label{Chapter1-125}\\n\\\\mathbf{y} = \
\\\\sum\\\\limits_{j = 1}^n {\\\\left\\\\langle {\\\\mathbf{c}_j^ \\\\dag,y} \
\\\\right\\\\rangle \\\\mathbf{c}_j}, \\n\\\\end{equation}\\nwhere \
$\\\\mathbf{c}_j$ are the columns of the $m \\\\times n$ matrix \
$\\\\mathbf{A}$\\n\\\\[\\n\\\\mathbf{A} \\\\equiv \\\\left[ {\\\\mathbf{c}_1 \
, \\\\ldots , \\\\mathbf{c}_n} \\\\right].\\n\\\\]\\nEquation \
\\\\eqref{Chapter1-125} expresses a reconstruction formula for a vector \
$\\\\mathbf{y}$ in terms of a sum over the product of a set of vectors \
$\\\\mathbf{c}_k$ (columns of the matrix $\\\\mathbf{A}$) multiplied by a set \
of scalar transform coefficients, obtained in the analysis stage by taking \
inner products of $\\\\mathbf{y}$ with the columns of the Hermitian conjugate \
of the right pseudo inverse, and represented by ${\\\\left\\\\langle \
{\\\\mathbf{c}_k ^ \\\\dag ,\\\\mathbf{y}} \\\\right \\\\rangle }$. \\n\\nA \
dual reconstruction formula to \\\\eqref{Chapter1-125} can be obtained \
starting with the identity\\n\\\\begin{equation}\\n\\\\label{Chapter1-126}\\n\
\\\\mathbf{y} = \\\\left( {\\\\mathbf{A} \\\\mathbf{A}^+} \\\\right)^{-1} \
(\\\\mathbf{A} \\\\mathbf{A}^+)\\\\mathbf{y} = ({\\\\mathbf{A}_R^\\\\dag})^+ \
{\\\\mathbf{A}}^+ \\\\mathbf{y}, ~~~ \\\\mathbf{y} \\\\in \\\\mathbb{C}^m ~,\
\\n\\\\end{equation}\\ninstead of \\\\eqref{Chapter1-123}. Now we \
have\\n\\\\begin{equation}\\n\\\\label{Chapter1-127}\\ny_{_k} = \
\\\\sum\\\\limits_{j = 1}^n {\\\\sum\\\\limits_{l = 1}^m { \
\\\\left[({\\\\mathbf{A}_R^ \\\\dag)^+} \\\\right]_{kj} \
(\\\\mathbf{A}^+)_{_{jl}}~ y_{_l} } }, ~~k = 1, \\\\ldots ,m, \
\\n\\\\end{equation} \\nwhile this time\\n\\\\[\\n\\\\sum\\\\limits_{l = \
1}^m {\\\\left[ {\\\\mathbf{A}^ +} \\\\right]_{jl} y_{_l}} = \
\\\\left\\\\langle {\\\\mathbf{c}_j,\\\\mathbf{y}} \
\\\\right\\\\rangle,\\n\\\\]\\nwhere $\\\\mathbf{c}_j$ are the columns of the \
matrix $\\\\mathbf{A}$, and we \
find\\n\\\\begin{equation}\\n\\\\label{Chapter1-128}\\n\\\\mathbf{y} = \
\\\\sum\\\\limits_{j = 1}^n {\\\\left\\\\langle {\\\\mathbf{c}_j,y} \\\\right\
\\\\rangle \\\\mathbf{c}_j ^ \\\\dag}, \\n\\\\end{equation}\\nwhere \
$\\\\mathbf{c}_j ^ \\\\dag$ are the columns of the $m \\\\times n$ matrix \
$(\\\\mathbf{A}_R^ \\\\dag)^ +$.\\n\\nThe reconstruction formula, however, is \
only useful when the right pseudo inverse matrix $\\\\mathbf{A}_R^\\\\dag$ \
can be computed in a stable fashion, which is possible only when the matrix $\
\\\\mathbf{A} \\\\mathbf{A}^ +$ has a reasonable condition \
number\\\\index{condition number}, that is, the magnitude of the ratio of the \
largest eigenvalue to the smallest eigenvalue is not very large \
\\\\footnote{Note that $\\\\mathbf{A} \\\\mathbf{A}^ +$ is Hermitian and \
positive semi definite and so has real and positive (or possibly zero) \
eigenvalues \\\\cite{GoVL96}. The same holds for $\\\\mathbf{A}^+ \
\\\\mathbf{A}$.}. But the eigenvalues of $\\\\mathbf{A} \\\\mathbf{A}^ +$ \
are the singular values\\\\index{singular values} of the matrix \
$\\\\mathbf{A}$ \\\\footnote{Any real or complex $m \\\\times n$ matrix \
$\\\\mathbf{A}$ can be factored as $\\\\mathbf{A} = \\\\mathbf{U} \\\\mathbf{\
\\\\Sigma} \\\\mathbf{V}^+$ where $\\\\mathbf{\\\\Sigma}$ is a $m \\\\times \
n$ real diagonal matrix of singular values $\\\\sigma_i \\\\ge 0$, $i=1, \
\\\\ldots ,k$ and $k = {\\\\rm Min}\\\\left( {m,n} \\\\right)$ \
\\\\cite{MoSt00}. When $m \\\\ne n$, $\\\\mathbf{\\\\Sigma}$ is constructed \
from the singular values by placing them along the main diagonal and then \
placing zeros in other locations to preserve the $m \\\\times n$ size of the \
matrix, which in such cases will inevitably result in some zero columns (when \
$m < n$), or some zero rows (when $m > n$). The singular values are the \
square roots of the intersection of the two sets of eigenvalues of the \
Hermitian and positive semi-definite matrices $\\\\mathbf{A} \\\\mathbf{A}^+$ \
and $\\\\mathbf{A}^+ \\\\mathbf{A}$. Assuming, without loss of generality, \
that $m < n$ and defining the square $m \\\\times m$ diagonal matrix \
$\\\\mathbf{\\\\Lambda}_m \\\\equiv {\\\\rm Diag}\\\\left[ {\\\\sigma _1^2 , \
\\\\ldots ,\\\\sigma _m^2 } \\\\right]$ and the square $n \\\\times n$ \
diagonal matrix $\\\\mathbf{\\\\Lambda}_n \\\\equiv {\\\\rm Diag}\\\\left[ {\
\\\\sigma _1^2 , \\\\ldots ,\\\\sigma _n^2 } \\\\right]$ we have \
$\\\\mathbf{A} \\\\mathbf{A}^+ \\\\mathbf{U} = \\\\mathbf{U} \
\\\\mathbf{\\\\Lambda}$ and $\\\\mathbf{A}^+ \\\\mathbf{A} \\\\mathbf{V} = \\\
\\mathbf{V} \\\\mathbf{\\\\Lambda}$. Thus, the eigenvectors of \
$\\\\mathbf{A} \\\\mathbf{A}^+$ are the columns of $\\\\mathbf{U}$, \
$\\\\mathbf{U} = \\\\left[ {\\\\mathbf{u}_1 , \\\\ldots ,\\\\mathbf{u}_n } \\\
\\right]$, while the eigenvectors of $\\\\mathbf{A}^+ \\\\mathbf{A}$ are the \
columns of $\\\\mathbf{V}$, $\\\\mathbf{V} = \\\\left[ {\\\\mathbf{v}_1 , \
\\\\ldots ,\\\\mathbf{v}_n } \\\\right]$. In addition, $\\\\mathbf{U} \
\\\\mathbf{U}^+ = \\\\mathbf{1}$, $\\\\mathbf{V} \\\\mathbf{V}^+ = \
\\\\mathbf{1}$. If the rank $r_{\\\\mathbf{A}} < m < n$ then $\\\\sigma _i \
> 0$, $1 \\\\le i \\\\le r_{\\\\mathbf{A}}$, and $\\\\sigma _i=0$, \
$r_{\\\\mathbf{A}} < i \\\\le m$ and we have the representation \
$\\\\mathbf{A} = \\\\sum\\\\limits_{i = 1}^{r_{\\\\mathbf{A}}} {\\\\sigma_i \
\\\\mathbf{u}_i \\\\mathbf{v}_i^ +}$ which can now be used to compute any \
function of a matrix, namely $f(\\\\mathbf{A}) = \\\\sum\\\\limits_{i = \
1}^{r_{\\\\mathbf{A}}} {f(\\\\sigma_i) \\\\mathbf{u}_i \\\\mathbf{v}_i^ \
+}$.}. Denoting the extreme singular values by $\\\\sigma_{min}$ and \
$\\\\sigma_{max}$ we have the following inequalities, \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-129}\\n\\\\sigma _{\\\\min } \
\\\\left\\\\| \\\\mathbf{y} \\\\right\\\\|^2 \\\\le \\\\left\\\\langle \
{\\\\mathbf{A}\\\\mathbf{A}^ + \\\\mathbf{y},\\\\mathbf{y}} \
\\\\right\\\\rangle \\\\le \\\\sigma _{\\\\max } \\\\left\\\\| \\\\mathbf{y} \
\\\\right\\\\|^2 , ~~ \\\\forall \\\\mathbf{y} \\\\in \\\\mathbb{C}^m ~.\\n\\\
\\end{equation}\\nUsing the reconstruction formula \\\\eqref{Chapter1-128} we \
have\\n\\\\[\\n\\\\left\\\\langle {\\\\mathbf{A} \\\\mathbf{A}^+ \
\\\\mathbf{y}, \\\\mathbf{y}} \\\\right\\\\rangle = \\\\left\\\\langle \
{\\\\mathbf{y},\\\\mathbf{A} \\\\mathbf{A}^+ \\\\mathbf{y}} \
\\\\right\\\\rangle = \\\\sum\\\\limits_{j = 1}^n {\\\\left\\\\langle \
{\\\\mathbf{c}_j ,y} \\\\right\\\\rangle \\\\left\\\\langle {y, \\\\mathbf{A} \
\\\\mathbf{A}^+ \\\\mathbf{c}_j^ \\\\dag} \\\\right\\\\rangle } \
~.\\n\\\\]\\nUsing the definition of the right pseudo inverse we have \
\\\\cite{CaMe79}\\n\\\\[\\n\\\\left( {\\\\mathbf{A} \\\\mathbf{A}^+ \
\\\\mathbf{c}_j^ \\\\dag} \\\\right)_k = \\\\left( {\\\\mathbf{A} \
\\\\mathbf{A}^+} \\\\right)_{ki} \\\\left[ {\\\\left( {\\\\mathbf{A}_R^ \
\\\\dag} \\\\right)^ +} \\\\right]_{ij} = \\\\left( {\\\\mathbf{A} \
\\\\mathbf{A}^+} \\\\right)_{ki} \\\\left[ {\\\\left( {\\\\mathbf{A} \
\\\\mathbf{A}^+} \\\\right)^{-1} } \\\\right]_{il} \\\\mathbf{A}_{lj} = \
\\\\mathbf{A}_{kj}~,\\n\\\\]\\nwhich is the $j$-th column of $\\\\mathbf{A}$ \
and so\\n\\\\[\\n\\\\left( {\\\\mathbf{A} \\\\mathbf{A}^+ \\\\mathbf{c}_j^ \\\
\\dag} \\\\right) = \\\\mathbf{c}_{j}~.\\n\\\\]\\nThe inequalities \
\\\\eqref{Chapter1-129} then \
become\\n\\\\begin{equation}\\n\\\\label{Chapter1-130}\\n\\\\sigma _{\\\\min \
} \\\\left\\\\| \\\\mathbf{y} \\\\right\\\\|^2 \\\\le \\\\left\\\\langle {\\\
\\mathbf{A}\\\\mathbf{A}^ + \\\\mathbf{y},\\\\mathbf{y}} \\\\right\\\\rangle \
= \\\\sum\\\\limits_{j = 1}^n {\\\\left| {\\\\left\\\\langle {\\\\mathbf{c}_j \
, \\\\mathbf{y}} \\\\right\\\\rangle } \\\\right|^2 } \\\\le \\\\sigma \
_{\\\\max } \\\\left\\\\| \\\\mathbf{y} \\\\right\\\\|^2 , ~~ \\\\forall \
\\\\mathbf{y} \\\\in \\\\mathbb{C}^m ~.\\n\\\\end{equation}\\n\\\\noindent \
These are also known as frame inequalities\\\\index{frame!inequalities}. Any \
set of $n$ vectors $\\\\left\\\\{ {\\\\mathbf{c}_k} \\\\right\\\\}$, \
satisfying the above inequalities is a frame\\\\index{frame} for \
$\\\\mathbb{C}^m$ and the quantities $\\\\sigma_{min}$ and $\\\\sigma_{max}$ \
are the associated frame bounds\\\\index{frame!bounds}. When \
$\\\\sigma_{min} = \\\\sigma_{max}$, i.e., the condition number of \
$\\\\mathbf{A}$ is $1$, the frame is tight\\\\index{frame!tight} and it \
mimics an orthogonal basis even though the frame vectors might be linearly \
dependent. If $\\\\sigma_{min} = \\\\sigma_{max}=1$ the frame becomes an \
orthogonal basis. Given non tight frame vectors $\\\\mathbf{c}_k$, a tight \
frame can be constructed by choosing \\n\\\\[\\n\\\\mathbf{c'}_k \\\\equiv \
\\\\left( {\\\\mathbf{A} \\\\mathbf{A}^+} \\\\right)^{{-1 \
\\\\mathord{\\\\left/ {\\\\vphantom {-1 2}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace} 2}} \\\\mathbf{c}_k ~,\\n\\\\]\\nwhere the \
square root operation is defined using the Singular Value Decomposition (SVD)\
\\\\index{singular values} of the associated matrix. The matrix \
$\\\\mathbf{A} \\\\mathbf{A}^+$ is the finite dimensional analogue of the \
basis operator ${T^+}_{\\\\phi} T_\\\\phi$ defined in a Hilbert space, \
\\\\eqref{Chapter1-101}. \\n\\n\\n\\\\section[Frames in $L_2 \\\\left( \
\\\\mathbb{R} \\\\right)$]{Frames in $L_2 \\\\left( \\\\mathbb{R} \
\\\\right)$}\\n\\\\label{FramesInInfinite}\\n\\nIn analogy to the previous \
section we generalize the concept of a Riesz basis\\\\index{basis!Riesz} in a \
Hilbert space by relaxing the exactness property of a Riesz basis (a Riesz \
basis is an exact frame\\\\index{frame!exact}) which allows for overcomplete \
(i.e., linearly dependent) sets that will lead to the concept of frames \
\\\\cite{Chri03}. We begin with the definition of a basis and an \
unconditional basis\\\\index{basis!unconditional} in $L_2(\\\\mathbb{R})$. \
\\n\\\\newtheorem{UnconditionalBasis}{Definition}\\n\\\\begin{defn}\\n\\\\\
label{UnconditionalBasis}\\nA set of functions $\\\\left\\\\{ {\\\\phi _n \
\\\\left( t \\\\right)} \\\\right\\\\}$, $n \\\\in \\\\mathbb{Z} $, whose \
closure is the entire space $L_2 \\\\left( \\\\mathbb{R} \\\\right)$ is a \
basis (or a Schauder basis\\\\index{basis!Schauder}) if every $x \\\\in L_2(\
\\\\mathbb{R})$ has the representation \\n\\\\[\\nx = \\\\sum\\\\limits_{n = \
- \\\\infty }^\\\\infty {c_n \\\\phi _n } ~,\\n\\\\]\\nwhere the \
coefficients $c_n$ are unique and depend on $x$. Furthermore, if, for this \
basis, two positive constants $A$ and $B$ exist such that\\n\\\\[\\nA\\\\left\
\\\\| \\\\underline{c} \\\\right\\\\|^2 \\\\le \\\\left\\\\| \
{\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {c_n \\\\phi _n } } \
\\\\right\\\\|^2 \\\\le B\\\\left\\\\| \\\\underline{c} \\\\right\\\\|^2 \\n\
\\\\]\\nfor every $\\\\underline{c} \\\\in l_2(\\\\mathbb{Z})$, then the \
basis is an unconditional basis.\\n\\\\end{defn}\\n\\nThe concept of a \
frame\\\\index{frame} can be introduced by relaxing the uniqueness of the \
expansion coefficients in an unconditional \
basis.\\n\\\\newtheorem{FrameDefinition}{Definition}\\n\\\\begin{defn}\\n\\\\\
label{FrameDefinition}\\nA set of functions $\\\\left\\\\{ {\\\\phi _n \
\\\\left( t \\\\right)} , n \\\\in \\\\mathbb{Z} \\\\right\\\\}$ is a frame \
in $L_2 \\\\left( \\\\mathbb{R} \\\\right)$ if positive constants $A$ and \
$B$, the frame bounds\\\\index{frame!bounds}, exist such that \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-131}\\nA\\\\left\\\\| x \\\\right\
\\\\|^2 \\\\le \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left| {\
\\\\left\\\\langle {\\\\phi _n ,x} \\\\right\\\\rangle } \\\\right|^2 } \
\\\\le B\\\\left\\\\| x \\\\right\\\\|^2 ~, \\n\\\\end{equation}\\nfor every \
$x \\\\in L_2(\\\\mathbb{R})$. If $A=B$ the frame is \
tight\\\\index{frame!tight}.\\n\\\\end{defn}\\n\\\\newtheorem{ExactFrameRiesz}\
{Theorem}\\n\\\\begin{thm}\\n\\\\label{ExactFrameRiesz}\\nReferring to \
definition \\\\ref{FrameDefinition} and equation \\\\eqref{Chapter1-131}, if \
the frame is exact then it is a Riesz basis. If $A=B=1$ then the frame is an \
orthonormal basis \\\\cite{Youn80, Daub92}.\\n\\\\end{thm}\\n\\\\noindent \\n\
\\\\noindent Thus, an othonormal basis\\\\index{basis!orthonormal} is exact \
and tight with unity frame bounds. Tight frames that are not orthonormal can \
be constructed by the union of two orthonormal bases \\n\\\\[\\n\\\\left\\\\{ \
{\\\\xi _n } \\\\right\\\\} = \\\\left\\\\{ {\\\\phi _n } \\\\right\\\\} \
\\\\cup \\\\left\\\\{ {\\\\psi _n } \\\\right\\\\}, ~ \\\\left\\\\langle {{\\\
\\xi _n }} \\\\mathrel{\\\\left | {\\\\vphantom {{\\\\xi _n } {\\\\xi _m \
}}}\\n \\\\right. \\\\kern-\\\\nulldelimiterspace} {{\\\\xi _m }} \\\\right\\\
\\rangle = \\\\left\\\\langle {{\\\\phi _n }} \\\\mathrel{\\\\left | \
{\\\\vphantom {{\\\\phi _n } {\\\\phi _m }}} \\\\right. \
\\\\kern-\\\\nulldelimiterspace}\\n {{\\\\phi _m }} \\\\right\\\\rangle = \\\
\\delta _{nm} \\n\\\\]\\nthat are not orthogonal to each other. The \
resulting frame is not orthogonal and for any $x(t)$ we have \
\\n\\\\[\\n\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left| \
{\\\\left\\\\langle {\\\\xi _n ,x} \\\\right\\\\rangle } \\\\right|^2 = \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left| \
{\\\\left\\\\langle {\\\\phi _n ,x} \\\\right\\\\rangle } \\\\right|^2 } + } \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left| \
{\\\\left\\\\langle {\\\\psi _n ,x} \\\\right\\\\rangle } \\\\right|^2 } = 2\
\\\\left\\\\| x \\\\right\\\\|^2 ~,\\n\\\\]\\nwhich clearly shows the \
corresponding frame bounds to be equal to $2$. If we choose the second basis \
to be an orthogonal but unnormalized set, e.g., $\\\\left\\\\| \\\\phi_n \
\\\\right\\\\|=2$, $\\\\forall ~ n$, then the resulting union is a frame that \
is neither tight, nor exact.\\n\\n\\nIn analogy to equations \
\\\\eqref{Chapter1-97} and \\\\eqref{Chapter1-98} we define the frame \
analysis operator $T_\\\\phi$ and its adjoint ${T^+}_\\\\phi$, the frame \
synthesis operator. Thus, we arrive at the definition of the frame \
operator\\\\index{frame!operator}.\\n\\\\newtheorem{FrameOperatorDeinition}{\
Definition}\\n\\\\begin{defn}\\n\\\\label{FrameOperatorDeinition}\\nGiven the \
frame functions $\\\\left\\\\{ {\\\\phi _n \\\\left( t \\\\right)} , n \\\\in \
\\\\mathbb{Z} \\\\right\\\\}$ we define the frame operator ${T^+}_\\\\phi \
T_\\\\phi$ by the \
equation\\n\\\\begin{equation}\\n\\\\label{Chapter1-132}\\n{T^+}_\\\\phi T_\\\
\\phi \\\\left\\\\{ {x\\\\left( t \\\\right)} \\\\right\\\\} = \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left\\\\langle {\\\\phi \
_n ,x} \\\\right\\\\rangle \\\\phi _n \\\\left( t \\\\right)}, ~ \\\\forall ~ \
x(t) \\\\in \
L_2(\\\\mathbb{R}).\\n\\\\end{equation}\\n\\\\end{defn}\\n\\\\noindent Using \
the definition of the frame operator and its adjoint, the frame definition \
equation \\\\eqref{Chapter1-131} can be rewritten as\\n\\\\begin{equation}\\n\
\\\\label{Chapter1-133}\\nA\\\\left\\\\| x \\\\right\\\\|^2 \\\\le \
\\\\left\\\\langle {x,{T^+}_\\\\phi T_\\\\phi x} \\\\right\\\\rangle \
\\\\equiv \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left| \
{\\\\left\\\\langle {\\\\phi _n ,x} \\\\right\\\\rangle } \\\\right|^2 } \
\\\\le B\\\\left\\\\| x \\\\right\\\\|^2 ~.\\n\\\\end{equation}\\nThe \
analysis operator $T_\\\\phi$ is not unitary since the frame elements are \
not, in general, orthonormal. The frame definition of equation \
\\\\eqref{Chapter1-133} implies that the frame operator ${T^+}_\\\\phi \
T_\\\\phi$ is bounded and positive definite, and hence invertible. The \
operator norms of ${T^+}_\\\\phi T_\\\\phi $ and its inverse are bounded by \
the frame bounds\\\\index{frame!bounds}. Using equation \
\\\\eqref{Chapter1-10} we have \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-134}\\n\\\\left\\\\| \
{{T^+}_\\\\phi T_\\\\phi } \\\\right\\\\| \\\\le B, ~~ \\\\left\\\\| \
{\\\\left( {{T^+}_\\\\phi T_\\\\phi } \\\\right)^{ - 1} } \\\\right\\\\| \
\\\\le A^{ - 1} ~.\\n\\\\end{equation}\\nThe ratio ${B \\\\mathord{\\\\left/ \
{\\\\vphantom {B A}} \\\\right. \\\\kern-\\\\nulldelimiterspace} A}$ is known \
as the frame redundancy ratio\\\\index{frame!redundancy ratio} and can be \
used as an approximation to the condition number\\\\index{condition number} \
of the operator ${T^+}_\\\\phi T_\\\\phi $. A finite frame ratio that is not \
too large and not too small is required for stable synthesis or recovery, \
i.e. reconstruction of a signal from its transform coefficients, as will be \
seen below in the frame representation equation. An orthonormal basis \
$\\\\hat \\\\phi _n \\\\left( t \\\\right)$ can be constructed for a frame \
with a reasonable frame ratio, using the square root of the inverse of the \
frame operator ${T^+}_\\\\phi T_\\\\phi $, \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-135}\\n\\\\hat \\\\phi _n \
\\\\left( t \\\\right) \\\\equiv \\\\left( {{T^+}_\\\\phi T_\\\\phi } \
\\\\right)^{ - {1 \\\\mathord{\\\\left/\\n {\\\\vphantom {1 2}} \\\\right. \\\
\\kern-\\\\nulldelimiterspace} 2}} \\\\phi _n \\\\left( t \\\\right) \
~.\\n\\\\end{equation}\\n\\\\newtheorem{DualFrameDefinition}{Definition}\\n\\\
\\begin{defn}\\n\\\\label{DualFrameDefinition}\\nGiven a frame $\\\\left\\\\{ \
\\\\phi_n, ~ n \\\\in \\\\mathbb{Z} \\\\right\\\\}$ with an invertible frame \
operator we define the dual frame\\\\index{frame!dual} elements \
by\\n\\\\begin{equation}\\n\\\\label{Chapter1-136}\\n\\\\xi _n \\\\left( t \\\
\\right) = \\\\left( {{T^+}_\\\\phi T_\\\\phi } \\\\right)^{-1} \\\\phi _n \
\\\\left( t \\\\right), ~ n \\\\in \
\\\\mathbb{Z}~.\\n\\\\end{equation}\\n\\\\end{defn}\\n\\\\noindent The \
inverse of this relation shows the linear dependence of the original frame \
elements (vectors), \
i.e.,\\n\\\\begin{equation}\\n\\\\label{Chapter1-137}\\n\\\\phi _n \\\\left( \
t \\\\right) = {T^+}_\\\\phi ~ T_\\\\phi ~ {\\\\xi _n (t)} = \
\\\\sum\\\\limits_{m = - \\\\infty }^\\\\infty {\\\\left\\\\langle {\\\\phi \
_m ,\\\\xi _n } \\\\right\\\\rangle \\\\phi _m \\\\left( t \\\\right)} \
~.\\n\\\\end{equation}\\n\\nThe dual frame is a new frame with frame bounds\\\
\\index{frame!bounds} $B^{ - 1}$ and $A^{ - 1}$, i.e., it satisfies the \
defining frame equation \\\\eqref{Chapter1-131} with the new bounds \
\\\\cite{Daub92},\\n\\\\begin{equation}\\n\\\\label{Chapter1-138}\\nB^{ - 1} \
\\\\left\\\\| x \\\\right\\\\|^2 \\\\le \\\\sum\\\\limits_{n = - \\\\infty \
}^\\\\infty {\\\\left| {\\\\left\\\\langle {\\\\xi _n ,x} \
\\\\right\\\\rangle } \\\\right|^2 } \\\\le A^{ - 1} \\\\left\\\\| x \
\\\\right\\\\|^2 ~.\\n\\\\end{equation}\\nThe frame representation for \
functions in $L_2 \\\\left( \\\\mathbb{R} \\\\right)$ is now expressed in the \
following dual frame reconstruction\\\\index{frame!dual reconstruction} \
equations \\n\\\\begin{equation}\\n\\\\label{Chapter1-139}\\nx\\\\left( t \
\\\\right) = \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty \
{\\\\left\\\\langle {\\\\phi _n ,x} \\\\right\\\\rangle \\\\xi _n \\\\left( t \
\\\\right)} = \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty \
{\\\\left\\\\langle {\\\\xi _n ,x} \\\\right\\\\rangle \\\\phi _n \\\\left( t \
\\\\right)} ~.\\n\\\\end{equation}\\nThe last equation recast in terms of \
frame operators is simply\\n\\\\begin{equation}\\n\\\\label{Chapter1-140}\\nx\
\\\\left( t \\\\right) = {T^+}_\\\\xi T_\\\\phi \\\\left\\\\{ {x\\\\left( t \
\\\\right)} \\\\right\\\\} = {T^+}_\\\\phi T_\\\\xi \\\\left\\\\{ \
{x\\\\left( t \\\\right)} \\\\right\\\\} ~.\\n\\\\end{equation}\\nThis \
equation is valid for all $x\\\\left( t \\\\right) \\\\in L_2 \\\\left( \
\\\\mathbb{R} \\\\right)$, and \
therefore\\n\\\\begin{equation}\\n\\\\label{Chapter1-141}\\n{T^+}_\\\\phi \
T_\\\\xi = {T^+}_\\\\xi T_\\\\phi = \\\\mathbf{1}\\n\\\\end{equation}\\nThe \
dual reconstruction equations \\\\eqref{Chapter1-139} are the solutions to \
two dual minimization problems as stated in the following theorem \
\\\\cite{Daub92}. \
\\n\\\\newtheorem{DualMinimumNormSolutions}{Theorem}\\n\\\\begin{thm}\\n\\\\\
label{DualMinimumNormSolutions}\\nThe coefficients $c_n \\\\in l_2 \
(\\\\mathbb{Z})$ in the expansion \\n\\\\[\\nx\\\\left( t \\\\right) = \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {c_n \\\\phi _n \\\\left( t \
\\\\right)}\\n\\\\]\\nthat minimize the square of the norm of the difference \
between the left and the right hand sides, are not unique. A minimum norm \
solution, i.e., a solution for which the quantity $\\\\sum\\\\limits_{n = - \
\\\\infty }^\\\\infty {\\\\left| {c_n } \\\\right|^2 }$ is a minimum among \
the set of all possible solutions, is given by $c_n = \\\\left\\\\langle {\\\
\\xi _n ,x} \\\\right\\\\rangle$. Similarly, the coefficients $c'_n \\\\in \
l_2 (\\\\mathbb{Z})$ that minimize the square of the norm of the difference \
between the two sides of the equation\\n\\\\[\\nx\\\\left( t \\\\right) = \
\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {c'_n \\\\xi_n \\\\left( t \
\\\\right)} \\n\\\\]\\nand have minimum norm is given by $c'_n = \
\\\\left\\\\langle {\\\\phi _n ,x} \\\\right\\\\rangle$.\\n\\\\end{thm}\\nThe \
reconstruction of a function from its coefficients via equations \
\\\\eqref{Chapter1-139} must be unique, i.e., the equality \
$\\\\left\\\\langle {\\\\phi _n ,x_1 } \\\\right\\\\rangle = \
\\\\left\\\\langle {\\\\phi _n ,x_2 } \\\\right\\\\rangle$ must imply that \
the two underlying functions are the same, $x_1(t) = x_2(t)$ where the \
equality is understood to mean $\\\\left\\\\| {x_1 - x_2 } \\\\right\\\\| = \
0$. Equivalently, we must \
have\\n\\\\begin{equation}\\n\\\\label{Chapter1-142}\\n\\\\left\\\\langle {\\\
\\phi _n ,e} \\\\right\\\\rangle = 0 \\\\Leftrightarrow e\\\\left( t \
\\\\right) = 0 ~.\\n\\\\end{equation}\\nThe frame inequalities \
\\\\eqref{Chapter1-131} and \\\\eqref{Chapter1-138} ensure that the above \
uniqueness condition is satisfied \\\\cite{Daub92}. \
\\n\\\\newtheorem{DualFrameReconstruction}{Theorem}\\n\\\\begin{thm}\\n\\\\\
label{DualFrameReconstruction}\\nA function $x(t)$ in $L_2(\\\\mathbb{R})$ \
can be uniquely reconstructed from the coefficients $\\\\left\\\\langle \
{\\\\phi _n ,x} \\\\right\\\\rangle$ or $\\\\left\\\\langle {\\\\xi _n ,x} \\\
\\right\\\\rangle$ if, and only if, the functions $\\\\phi_n(t)$ and \
$\\\\xi_n(t)$ form dual frames.\\n\\\\end{thm}\\n\\n\\nThe following theorem \
shows the close connection between a frame and a reproducing \
kernel\\\\index{reproducing kernel} in a reproducing kernel Hilbert \
space\\\\index{reproducing kernel!Hilbert space} \\\\cite{Daub92, \
BeAg04}.\\n\\\\newtheorem{DualFrameKernel}{Theorem}\\n\\\\begin{thm}\\n\\\\\
label{DualFrameKernel}\\nA Hilbert space with a frame and frame bounds that \
allow the construction of dual frames $\\\\left\\\\{ {\\\\phi _n ,\\\\xi _n } \
\\\\right\\\\}$, admits a unique reproducing kernel that has the following \
representation\\n\\\\begin{equation}\\n\\\\label{Chapter1-143}\\nK\\\\left( \
{t,t'} \\\\right) = \\\\sum\\\\limits_{n =-\\\\infty }^\\\\infty {\\\\xi_n^* \
\\\\left(t \\\\right)\\\\phi_n \\\\left(t' \\\\right)} ~,\\n\\\\end{equation}\
\\nand that is self adjoint, i.e., $K\\\\left( {t,t'} \\\\right) = K^ * \
\\\\left( {t',t} \\\\right)$.\\n\\\\end{thm}\\n\\\\noindent Equation \
\\\\eqref{Chapter1-41} is a special case of this result applied to an \
orthonormal basis.\\n\\n\\n\\\\section[Dual Frames]{Dual Frame Construction \
Algorithm}\\n\\nThe dual frame\\\\index{frame!dual} reconstruction formulae \
expressed in equation \\\\eqref{Chapter1-139} can only be used when the dual \
frame elements are known. If only the frame elements are available then an \
iterative scheme can be used to approximate the dual frame elements. The \
approximation is useful only when the frame ratio ${B \\\\mathord{\\\\left/ {\
\\\\vphantom {B A}} \\\\right. \\\\kern-\\\\nulldelimiterspace} A}$ is not \
too much larger than $1$.\\n\\nUsing the definition of the dual frame in \
equation \\\\eqref{Chapter1-136} the reconstruction formula \
\\\\eqref{Chapter1-139} can be written as\\n\\\\[\\nx\\\\left( t \\\\right) = \
\\\\sum\\\\limits_{n=-\\\\infty }^\\\\infty {\\\\left\\\\langle {\\\\phi_n \
,x} \\\\right\\\\rangle} \\\\left( {{T^+}_\\\\phi T_\\\\phi} \\\\right)^{-1} \
\\\\phi_n \\\\left( t \\\\right)\\n\\\\]\\nwhich is equivalent to \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-144}\\n\\\\left( {1 - \
\\\\frac{2}{{A + B}}{T^+}_\\\\phi T_\\\\phi } \\\\right) x\\\\left( t \
\\\\right) = x\\\\left( t \\\\right) - \\\\frac{2}{{A + \
B}}\\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left\\\\langle \
{\\\\phi _n ,x} \\\\right\\\\rangle \\\\phi _n \\\\left( t \\\\right)} \
~.\\n\\\\end{equation}\\nIt can be shown that \\\\cite{Daub92} \\n\\\
\\begin{equation}\\n\\\\label{Chapter1-145}\\n\\\\left\\\\| {1 - \
\\\\frac{2}{{A + B}}{T^+}_\\\\phi T_\\\\phi} \\\\right\\\\| \\\\le \
\\\\frac{{B - A}}{{B + A}} < 1 ~,\\n\\\\end{equation}\\nand so according to \
theorem \\\\ref{NeumannExpansion} and equation \\\\eqref{Chapter1-13} the \
operator \\n\\\\[\\n\\\\left({1-\\\\left({1-\\\\frac{2}{{A + B}}{T^+}_\\\\phi \
T_\\\\phi} \\\\right)} \\\\right)^{-1} = \\\\frac{{A + B}}{2}\\\\left({{T^+}_\
\\\\phi T_\\\\phi} \\\\right)^{-1} \\n\\\\]\\nhas a convergent Neumann \
expansion\\\\index{Neumann expansion}. \
Thus,\\n\\\\begin{equation}\\n\\\\label{Chapter1-146}\\n\\\\left( \
{{T^+}_\\\\phi T_\\\\phi} \\\\right)^{-1} = \\\\frac{2}{{A + \
B}}\\\\sum\\\\limits_{k = 0}^\\\\infty {\\\\left( {1 - \\\\frac{2}{{A + \
B}}{T^+}_\\\\phi T_\\\\phi} \\\\right)} ^k ~.\\n\\\\end{equation}\\nAn \
approximation to the dual frame vectors is obtained by using the first $N$ \
terms of the Neumann series in equation \\\\eqref{Chapter1-136}. Thus, \\n\\\
\\begin{equation}\\n\\\\label{Chapter1-147}\\n\\\\xi _{_{nN}} \\\\left( t \
\\\\right) \\\\equiv \\\\frac{2}{{A + B}}\\\\sum\\\\limits_{k = 0}^{N-1} \
{\\\\left( {1 - \\\\frac{2}{{A + B}}{T^+}_\\\\phi T_\\\\phi } \\\\right)} \
^k \\\\phi _n \\\\left( t \\\\right) ~.\\n\\\\end{equation}\\nThe \
approximated dual frame when used in equation \\\\eqref{Chapter1-140} gives \
the approximate reconstructed signal. The norm of the error in the \
reconstruction can be shown to decay \
exponentially,\\n\\\\begin{equation}\\n\\\\label{Chapter1-148}\\n\\\\left\\\\|\
{x - \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {\\\\left\\\\langle \
{\\\\phi _n ,x} \\\\right\\\\rangle \\\\xi _{_{nN}} } } \\\\right\\\\| \\\\le \
\\\\left( {\\\\frac{{B - A}}{{B + A}}} \\\\right)^{N} \\\\left\\\\| x \
\\\\right\\\\| ~.\\n\\\\end{equation}\\n\\nFrames provide a powerful method \
of signal representation because of their redundancy. Among important \
advantages of a redundant basis are better approximations of the continuous \
wavelet transform coefficients, robustness to quantization effects and fewer \
restrictions in the choice of analyzing basis functions which makes it easier \
to design functions matching specific signal characteristics. In addition, a \
signal can be successfully reconstructed even when some coefficients are \
missing. However, frames can be difficult to use for signal reconstruction \
since the reconstruction formula requires the computation of the dual frame \
elements, which as has been noted above, is a cumbersome and approximate \
process. Orthonormal basis functions (tight frames with bounds $A=B=1$), on \
the other hand, are far easier to deal with. For wavelets of compact support \
we will study both orthonormal and biorthogonal basis functions, since as we \
shall see later, the transform coefficients can be calculated using finite \
impulse response (FIR) filters, and without the need to compute any of the \
basis functions.\\n\\nLocalization of signals both in time and frequency, or \
in time and scale are major motivations for studying the windowed Fourier \
transform and the continuous wavelet transform: both are linear maps between \
$L_2 \\\\left( \\\\mathbb{R} \\\\right)$ and $L_2 \\\\left( {\\\\mathbb{R}}^2 \
\\\\right)$ and both transforms use two continuous parameters: time shift and \
frequency for the windowed Fourier transform, and time shift and scale for \
the continuous wavelet transform. The property that is common to both is \
that the transform coefficients for a function $x(t)$ are obtained by taking \
the inner product between $x(t)$ and a family of analyzing functions that are \
constructed from the same mother window, in the case of the windowed Fourier \
transform, and the same mother wavelet, in the case of the continuous wavelet \
transform. \\n\\nThe process of building frames and orthonormal bases in \
either case is based on discretization of the continuous variables on which \
the analyzing functions depend. This results in analyzing functions that \
carry two integer indices: one for time translation and one for frequency, in \
the case of the windowed Fourier transform, and one for time translation and \
one for scale, in the continuous wavelet transform case. In either case we \
may represent the basis functions by $\\\\xi _{mn} \\\\left( t \\\\right)$ \
and the corresponding inner products by $c_{mn}$ with the following analysis \
and synthesis (reconstruction) equations \
\\n\\\\begin{equation}\\n\\\\label{Chapter1-149}\\n~c_{mn} = \
\\\\left\\\\langle {\\\\xi _{mn} ,x} \\\\right\\\\rangle ,~x\\\\left( t \
\\\\right) = \\\\sum\\\\limits_{n = - \\\\infty }^\\\\infty {c_{mn} } \
\\\\xi _{mn} \\\\left( t \\\\right),\\n\\\\end{equation}\\nand, in the Dirac \
notation and summation over repeated indices, the completeness \
relation\\n\\\\begin{equation}\\n\\\\label{Chapter1-150}\\n\\\\left| {\\\\xi \
_{mn} } \\\\right\\\\rangle \\\\left\\\\langle {\\\\xi _{mn} } \\\\right| = \
\\\\mathbf{1} ~.\\n\\\\end{equation}\\nIf the associated frames are actually \
orthonormal bases then we will have the additional orthonormality \
relations\\n\\\\begin{equation}\\n\\\\label{Chapter1-151}\\n\\\\left\\\\\
langle {{\\\\xi _{mn} }}\\n \\\\mathrel{\\\\left | {\\\\vphantom {{\\\\xi \
_{mn} } {\\\\xi _{lk} }}}\\n \\\\right. \\\\kern-\\\\nulldelimiterspace}\\n \
{{\\\\xi _{lk} }} \\\\right\\\\rangle = \\\\delta _{ml} \\\\delta _{nk} \
~.\\n\\\\end{equation}\\n\\n\\n%\\\\clearpage\\n\\\\section{Exercises}\\n\\n\\\
n\\n\\\\paragraph{1}\\n\\nConsider the set of functions $\\\\left\\\\{ \
{1,t,t^2 , \\\\ldots ,t^{N - 1} } \\\\right\\\\}$ defined on the closed \
interval $\\\\left[ { - 1,1} \\\\right]$. The members of this set are $N$ \
linearly independent functions. Using the inner product $\\\\left\\\\langle \
{f,g} \\\\right\\\\rangle \\\\equiv \\\\int\\\\limits_{ - 1}^1 {f\\\\left( t \
\\\\right)} g\\\\left( t \\\\right)dt$ and the Gram-Schmidt procedure find a \
set of $N$ orthonormal polynomials (the latter are known as the Legendre \
polynomials).\\n\\n\\n\\\\paragraph{2}\\n\\nRepeat the last problem using the \
same set of functions, but this time use the following inner \
product\\n$\\\\left\\\\langle {f,g} \\\\right\\\\rangle \\\\equiv \
\\\\int\\\\limits_{ - 1}^1 {f\\\\left( t \\\\right)} \\\\left( {1 - t^2 } \
\\\\right)^{ - \\\\frac{1}{2}} g\\\\left( t \\\\right)dt$. The resulting \
orthonormal functions are known as the Chebyshev polynomials $T_0 \\\\left( t \
\\\\right) \\\\equiv \\\\pi ^{ - \\\\frac{1}{2}}$, and $T_n \\\\left( t \
\\\\right) \\\\equiv \\\\left( {\\\\frac{\\\\pi }{2}} \\\\right)^{ - \
\\\\frac{1}{2}} \\\\cos \\\\left[ {n\\\\cos ^{ - 1} \\\\left( t \\\\right)} \
\\\\right]$, $n = 1,2, \\\\ldots$.\\n\\n\\\\paragraph{3}\\n\\nProve equation \
\\\\eqref{Chapter1-9}.\\n\\n\\n\\n\\\\paragraph{4}\\n\\nProve the \
completeness relation \\\\eqref{Chapter1-114}.\\n\\n\\n\\n\\\\paragraph{5}\\n\
\\nConsider the function defined to be $e^{ - \\\\left| n \\\\right|}$ on \
intervals $[n, n+1]$, $n \\\\in \\\\mathbb{Z}$. Show that this function is \
square integrable and is in the subspace \
$\\\\mathscr{V}_0$.\\n\\n\\n\\\\paragraph{6}\\n\\nConsider a continuous time \
function $f\\\\left( t \\\\right) = \\\\sum\\\\limits_{n = - \\\\infty \
}^\\\\infty {h\\\\left[ n \\\\right]} \\\\phi \\\\left( {t - n} \\\\right)$. \
Calculate the Fourier transform of the function expressing your result in \
terms of the Fourier transforms $H\\\\left( \\\\omega \\\\right)$ and \
$\\\\Phi \\\\left( \\\\omega \
\\\\right)$.\\n\\n\\n\\\\paragraph{7}\\n\\nProve the result \
\\n\\\\[\\n\\\\int\\\\limits_{ - \\\\infty }^\\\\infty {\\\\frac{{\\\\sin \\\
\\left( {a\\\\pi t} \\\\right)}}{{a\\\\pi t}}e^{ - i\\\\omega t} dt = \
\\\\left\\\\{ \\\\begin{array}{l}\\n {1 \\\\mathord{\\\\left/ {\\\\vphantom \
{1 a}} \\\\right. \\\\kern-\\\\nulldelimiterspace} a}, - a\\\\pi < \\\\omega \
< a\\\\pi \\\\\\\\ \\n 0, ~~~~~~~ {\\\\rm otherwise} ~, \\\\\\\\ \\n \
\\\\end{array} \\\\right.} \\n\\\\]\\nby calculating the inverse Fourier \
transform of the right hand side. Use this result to find the Fourier \
transform of $p_k\\\\left(t \\\\right)$ as defined in equations \
\\\\eqref{Chapter1-92} and \\\\eqref{Chapter1-89} \
\\\\cite{MoSt00}.\\n\\n\\\\paragraph{8}\\n\\nGiven a signal $s\\\\left( t \
\\\\right)$ whose band is limited to $\\\\left[ { - B, + B} \\\\right]$, we \
use the completeness of the functions $p_k \\\\left( t \\\\right)$ (see the \
last problem and equations \\\\eqref{Chapter1-92} and \\\\eqref{Chapter1-89}) \
to write $s\\\\left( t \\\\right) = \\\\sum\\\\limits_{k = - \\\\infty \
}^\\\\infty {c_k p_k \\\\left( t \\\\right)}$. Using the orthonormality of \
the functions $\\\\left\\\\{ {p_k \\\\left( t \\\\right)} \\\\right\\\\}$ we \
find the coefficients $c_k = \\\\int\\\\limits_{ - \\\\infty }^\\\\infty {s\
\\\\left( t \\\\right)p_k \\\\left( t \\\\right)} dt$. Use theorem \
\\\\ref{FTParseval} and the Fourier transform result of the previous problem \
to calculate this coefficient. This is a proof of the sampling \
theorem\\\\index{sampling theorem} (theorem \
\\\\ref{SamplingTheorem}).\\n\\n\\\\paragraph{9}\\nUsing the reproducing \
kernel\\\\index{reproducing kernel} function of equation \
\\\\eqref{Chapter1-85}, show that the kernel is a projection operator from \
$L_2 \\\\left( \\\\mathbb{R} \\\\right)$ into $L_2^\\\\Omega \\\\left( \
\\\\mathbb{R} \
\\\\right)$.\\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\
%%%%%%%%%%%%%%%%%%%%%%%%%%%\\n%\\n% END \
OF CHAPTER \
1\\n%\\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\
%%%%%%%%%%%%%%%%%%%%\\n\\n\\n\\n\\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\\n%\\n% \
CHAPTER \
2\\n%\\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\
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