Essential Wavelets for Statistical Applications and Data Analysis [Softcover reprint of the original 1st ed. 1997] 9781461268765, 9781461207092, 1461268761

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To Christine

R. Todd Ogden

Essential Wavelets for Statistical Applications and Data Analysis

Springer Science+Business Media, LLC

R. Todd Ogden Department of Statisties University of South Caro lina Columbia, SC 29208

Library of Congress Cataloging-in-Publication Data Ogden, R. Todd, 1965Essential wavelets for statistic al applieations and data analysis I R. Todd Ogden. p. em. Includes bibliographieal referenees (p. 191-198) and index. ISBN 978-1-4612-6876-5 ISBN 978-1-4612-0709-2 (eBook) DOI 10.1007/978-1-4612-0709-2 1. Wavelets (Mathematies) 2. Mathernatieal statisties 1. Title. QA403.3.043 1997 519.5--de20 97-27379 CIP

Printed on acid-free paper © 1997 Springer Science+Business Media New York Originally published by Birkhăuser Boston in 1997 Softeover reprint ofthe hardeover Ist edition 1997 Copyright is not claimed for works of U.S. Government employees. AII rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific c1ients is granted by Birkhăuser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided thatthe base fee of$6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC.

ISBN 978-1-4612-6876-5 Typeset in Jb.Tp,X by ShadeTree Designs, Minneapolis, MN. Cover design by Spencer Ladd, Somerville, MA.

Contents Preface Prologue: Why Wavelets? 1 Wavelets: A Brief Introduction 1.1 The Discrete Fourier Transform 1.2 The Haar System Multiresolution Analysis The Wavelet Representation Goals of Multiresolution Analysis 1.3 Smoother Wavelet Bases

2 Basic Smoothing Techniques 2.1 Density Estimation

2.2 2.3

Histograms Kernel Estimation Orthogonal Series Estimation Estimation of a Regression Function Kernel Regression Orthogonal Series Estimation Kernel Representation of Orthogonal Series Estimators

3 Elementary Statistical Applications 3.1 Density Estimation Haar-Based Histograms Estimation with Smoother Wavelets 3.2 Nonparametric Regression

4 Wavelet Features and Examples 4.1 Wavelet Decomposition and Reconstruction Two-Scale Relationships The Decomposition Algorithm The Reconstruction Algorithm 4.2 The Filter Representation

ix xiii

1

1

7

14 16 22 23 29 29 31 32 35 38 39 42 45 49 49 49 52 54 59 59 60 62 63 66

vi

4.3 Time-Frequency Localization The Continuous Fourier Transform The~mdowedFourierTranMorm

The Continuous Wavelet Transform 4.4 Examples of Wavelets and Their Constructions Orthogonal Wavelets Biorthogonal Wavelets Semiorthogonal Wavelets

69 69 72

74 79 81 83 87

5 Wavelet-based Diagnostics 5.1 Multiresolution Plots 5.2 Time-Scale Plots 5.3 Plotting Wavelet Coefficients 5.4 Other Plots for Data Analysis

89 89 92 95 100

6 Some Practical Issues 6.1 The Discrete Fourier Transform of Data

103 104 104 105 107 110 111 112 113 114 115

The Fourier Transform of Sampled Signals The Fast Fourier Transform 6.2 The Wavelet Transform of Data 6.3 Wavelets on an Interval Periodic Boundary Handling Symmetric and Antisymmetric Boundary Handling Meyer Boundary Wavelets Orthogonal Wavelets on the Interval 6.4 When the Sample Size is Not a Power of Two

7 Other Applications 7.1 Selective Wavelet Reconstruction Wavelet Thresholding Spatial Adaptivity Global Thresholding Estimation of the Noise Level 7.2 More Density Estimation 7.3 Spectral Density Estimation 7.4 Detections of Jumps and Cusps

119 119 124 126 128 131 132 133 140

8 Data Adaptive Wavelet Thresholding 8.1 SURE Thresholding 8.2 Threshold Selection by Hypothesis Testing

143 144 149 151 154 156 161

Recursive Testing Minimizing False Discovery 8.3 Cross-Validation Methods 8.4 Bayesian Methods

vii

9 Generalizations and Extensions 9.1 Two-Dimensional Wavelets 9.2 Wavelet Packets

167

Wavelet Packet Functions The Best Basis Algorithm 9.3 Translation Invariant Wavelet Smoothing

167 173 174 177 180

Appendix

185

References

191

Glossary of Notation

199

Glossary of Terms

201

Index

205

Preface I once heard the book by Meyer (1993) described as a "vulgarization" of wavelets. While this is true in one sense of the word, that of making a subject popular (Meyer's book is one of the early works written with the nonspecialist in mind), the implication seems to be that such an attempt somehow cheapens or coarsens the subject. I have to disagree that popularity goes hand-in-hand with debasement. While there is certainly a beautiful theory underlying wavelet analysis, there is plenty of beauty left over for the applications of wavelet methods. This book is also written for the non-specialist, and therefore its main thrust is toward wavelet applications. Enough theory is given to help the reader gain a basic understanding of how wavelets work in practice, but much of the theory can be presented using only a basic level of mathematics. Only one theorem is formally stated in this book, with only one proof. And these are only included to introduce some key concepts in a natural way.

Aim and Scope This book was written to become what the reference that I wanted when I began my own study of wavelets. I had books and papers, I studied theorems and proofs, but no single one of these sources by itself answered the specific questions I had: In order to apply wavelets successfully, what do I need to know? And why do I need to know it? It is my hope that this book will answer these questions for others in the same situation. In keeping with the title of this book, I have attempted to pare down the possible number of topics of coverage to just the essentials required for statistical applications and analysis of data. New statistical applications are being developed quickly, so due to the combination of careful choosing of topics and natural delays in writing and printing, this book is necessarily incomplete. It is hoped, however, that the introduction provided in this text will provide a suitable foundation for readers to jump off into other wavelet-related topics. I am of the opinion that basic wavelet methods of smoothing functions, for example, should be as widely understood as standard kernel methods are now. Admittedly, understanding wavelet methods requires a substantial amount of overhead, in terms of time and effort, but the richness of wavelet

x

PREFACE

applications makes such an investment well worth it. This modest work is thus put forward to widen the circle of wavelet literacy. It is important to point out that I am not at all advocating the complete abandonment of all other methods. In a recent article, Fan, et at. (1996) discuss local versions of some standard smoothing techniques and show that they provide a good alternative to wavelet methods, and in fact may be preferred in many applications because of their familiarity. This book was written primarily to increase the familiarity of wavelets in data analysis: wavelets are Simply another useful tool in the toolbag of applied statisticians and data analysts. The treatment of topics in this book assumes only that the reader is familiar with calculus and linear algebra, with a basic understanding of elementary statistical theory. With this background, this book is essentially self-contained, with other topics (Fourier analysis, L2 function space, function estimation, etc.) treated when introduced. A brief overview of L2 function space is given as an appendix, along with glossaries of notation and terms. Thus, the material is accessible to a wide audience, including graduate students and advanced undergraduates in mathematics and statistics, as well as those in other disciplines interested in data analysis. Mathematically sophisticated readers can use this reference as quick reading to gain a basic understanding of how wavelets can be used.

Chapter Synopses The Prologue gives a basic overview of the topic of wavelets and describes their most important features in nonmathematicallanguage. Chapter 1 provides a fundamental introduction to what wavelets are, with brief hints as to how they can be used in practice. Though the results of this chapter apply to general orthogonal wavelets, the material is presented primarily in terms of the simplest case of wavelet: the Haar basis. This greatly simplifies the treatment in introducing wavelet features, and once the basic Haar framework is understood, the ideas are readily extended to smoother wavelet bases. Leaving the treatment of wavelets momentarily, Chapter 2 gives a general introduction to fundamental methods of statistical function estimation in such a way that will lead naturally to basic applications of wavelets. This will of course be review material for readers already familiar with kernel and orthogonal series methods; it is included primarily for the non-specialist. Chapter 3 treats the wavelet versions of the smoothing methods described in Chapter 2, applied to density estimation and nonparametric regression. Chapter 4 returns to describing wavelets, continuing the coverage of Chapter 1. It covers more details of the earlier introduction to wavelets, and treats wavelets in more generality, introducing some of the fundamental properties of wavelet methods: algorithms, filtering, wavelet extension of the Fourier transform, and examples of wavelet families. This chapter is not,

Preface

xi

strictly speaking, essential for applying wavelet methods, but it provides the reader with a better understanding of the principles that make wavelets work well in practice. Chapters 6-9 deal with applying wavelet methods to various statistical problems. Chapter 5 describes diagnostic methods essential to a complete data analysis. Chapter 6 discusses the important practical issues that arise in wavelet analysis of real data. Chapter 7 extends and enhances the basic wavelet methods of Chapter 3. Chapter 8 gives an overview of current research in data dependent wavelet threshold selection. Finally, Chapter 9 provides a basic background into wavelet-related methods which are not explicitly treated in earlier chapters. The information in this book could have been arranged in a variety of orders. If it were intended strictly as a reference book, a natural way to order the information might be to place the chapters dealing primarily with the mathematics of wavelets (Chapters 2, 5, and 10) at the beginning, followed by the statistical application chapters (Chapters 4, 8, and 9), with the diagnostic chapter last, the smoothing chapter being included as an appendix. Instructors using this book in a classroom might cover the topics roughly in the order given, but with the miscellaneous topics in Chapter 4 distributed strategically within subsequent applications chapters. The current order was carefully selected so as to provide a natural path through wavelet introduction and application to facilitate the reader's first learning of the subject, but with like topics grouped sufficiently close together so that the book will have some value for subsequent reference.

Supplements on the World Wide Web The figures in this book were mostly generated using the commercial S-Plus software package, some using the S-Plus Wavelet Toolkit, and some using the freely available set of S-Plus wavelet subroutines by Guy Nason, available through StatIlb (http://l ib. s ta t . emu. edu I). To encourage readers' experimentation with wavelet methods and facilitate other applications, I have made available the S-Plus functions for generating most of the pictures in this book over the World Wide Web (this is in lieu of including source code in the text). These will be located both on Birkhiiuser's web site (http://www.birkhauser.eom/books/isbn/O-8176-3864-4/), and as a link from my personal home page (http://www.stat.se . edu/ - ogdenl), which will also contain errata and other information regarding this book. As they become available, new routines for wavelet-based analysis will be included on these pages as well. Though I have only used the S-Plus software, there are many other available software packages available, such as WaveLab, an extensive collection of MATLAB-based routines for wavelet analysis which is available free from Stanford's Statistics Department WWW site. Vast amounts of wavelet-related material is available through the www,

xii

PREFACE

including technical reports, a wavelet newsletter, Java applets, lecture notes, and other forms of information. The web pages for this book, which will be updated periodically, will also describe and link relevant information sites.

Acknowledgments This book represents the combination of efforts of many different people, some of whom I will acknowledge here. Thanks are due to Manny Parzen and Charles Chui for their kind words of encouragement at the outset of this project. I gratefully acknowledge Andrew Bruce, Hong-Ye Gao and others at StatSci for making available their S-PLUS Wavelet software. The suggestions and comments by Jon Buckheit, Christian Cenker, Cheng Cheng, and Webster West were invaluable in improving the presentation of the book and correcting numerous errors. I am deeply indebted to each of them. Mike Hilton and Wim Sweldens have the ability to explain difficult concepts in an easily understandable way-my writing of this book has been motivated by their examples in this regard. Carolyn Artin read the entire manuscript and made countless excellent suggestions on grammar and wording. Joe Padgett, John Spurrier, Jim Lynch, and my other colleagues at the University of South Carolina have been immensely supportive and helpful; I thank them as well. Thanks are also due to Wayne Yuhasz and Lauren Lavery at Birkhiiuser for their support and encouragement of the project. Finally, my deepest thanks go to my family: my wife Christine and daughter Caroline, who stood beside me every word of the way.

PROLOGUE

Why Wavelets? The development of wavelets is fairly recent in applied mathematics, but wavelets have already had a remarkable impact. A lot of people are now applying wavelets to a lot of situations, and all seem to report favorable results. What is it about wavelets that make them so popular? What is it that makes them so useful? This prologue will present an overview in broad strokes (using descriptions and analogies in lieu of mathematical formulas). It is intended to be a brief preview of topics to be covered in more detail in the chapters. It might be useful for the reader to refer back to the prologue from time to time, to prevent the possibility of getting bogged down in mathematical detail to the extent that the big picture is lost. The prologue describes the forest; the trees are the subjects of the chapters. Broadly defined, a wavelet is simply a wavy function carefully constructed so as to have certain mathematical properties. An entire set of wavelets is constructed from a single "mother wavelet" function, and this set provides useful "building block" functions that can be used to describe any in a large class of functions. Several different possibilities for mother wavelet functions have been developed, each with its associated advantages and disadvantages. In applying wavelets, one only has to choose one of the available wavelet families; it is never necessary to construct new wavelets from scratch, so there is little emphasis placed on construction of specific wavelets. Roughly speaking, wavelet analysis is a refinement of Fourier analysis. The Fourier transform is a method of describing an input signal (or function) in terms of its frequency components. Consider a simple musical analogy, following Meyer (1993) and others. Suppose someone were to play a sustained three-note chord on an organ. The Fourier transform of the resulting digitized acoustic signal would be able to pick out the exact frequencies of the three component notes, and the chord could be analyzed by studying the relationships among the frequencies. Suppose the organist plays the same chord for a measure, then abruptly change to a different chord and sustains that for another measure. Here, the classical Fourier analysis becomes confused. It is able to determine the frequencies of all the notes in either chord, but it is unable to distinguish which frequencies belong to the first chord and which are part of the second. Essentially, the frequencies are averaged over the two measures, and the

xiv

WHY WAVELETS?

Fourier reconstruction would sound all frequencies simultaneously, possibly sounding quite dissonant. While usual Fourier methods do a very good job at picking out frequencies from a signal consisting of many frequencies, they are utterly incapable of dealing properly with a signal that is changing over time. This fact has been well-known for years. To increase the applicability of Fourier analysis, various methods such as "windowed Fourier transforms" have been developed to adapt the usual Fourier methods to allow analysis of the frequency content of a signal at each time. While some success has been achieved, these adaptations to the Fourier methods are not completely satisfactory. Wmdowed transforms can localize simultaneously in time and in frequency, but the amount of localization in each dimension remains fixed. With wavelets, the amount of localization in time and in frequency is automatically adapted, in that only a narrow time-window is needed to examine high-frequency content, but a wide time-window is allowed when investigating low-frequency components. This good time-frequency localization is perhaps the most important advantage that wavelets have over other methods. It might not be immediately clear, however, how this time-frequency localization is helpful in statistics. In statistical function estimation, standard methods (e.g., kernel smoothers or orthogonal series methods) rely upon certain assumptions about the smoothness of the function being estimated. With wavelets, such assumptions are relaxed conSiderably. wavelets have a built-in "spatial adaptivity" that allows efficient estimation of functions with discontinuities in derivatives, sharp spikes, and discontinuities in the function itself. Thus, wavelet methods are useful in nonparametric regression for a much broader class of functions. Wavelets are intrinsically connected to the notion of "multiresolution analysis." That is, objects (signals, functions, data) can be examined using widely varying levels of focus. As a simple analogy, consider looking at a house. The observation can be made from a great distance, at which the viewer can discern only the basic shape of the structure-the pitch of the roof, whether or not it has an attached garage, etc. As the observer moves closer to the building, various other features of the house come into focus. One can now count the number of windows and see where the doors are located. Moving closer still, even smaller features come into clear view: the house number, the pattern on the curtains. Continuing, it is possible even to examine the pattern of the wood grain on the front door. The basic framework of all these views is essentially the same using wavelets. This capability of multiresolution analysis is known as the "zoom-in, zoom-out" property. Thus, frequency analysis using the Fourier decomposition becomes "scale analysis" using wavelets. This means that it is possible to examine features of the signal (the function, the house) of any size by adjusting a scaling parameter in the analysis. Wavelets are regarded by many as primarily a new subject in pure mathe-

Why Wavelets?

xv

matics. Indeed, many papers published on wavelets contain esoteric-looking theorems with complicated proofs. This type of paper might scare away people who are primarily interested in applications, but the vitality of wavelets lies in their applications and the diversity of these applications. The objective of this book is to introduce wavelets with an eye toward data analysis, giving only the mathematics necessary for a good understanding of how wavelets work and a knowledge of how to apply them. Since no wavelet application exists in complete isolation (in the sense that substantial overlap can be found among virtually all applications), we review here some of the ways wavelets have been applied in various fields and consider how specific advantages of wavelets in these fields can be exploited in statistical analysis as well. Certainly, wavelets have an "interdisciplinary" flavor. Much of the predevelopment of the foundations of what is now known as wavelet analysis was led by Yves Meyer, Jean Morlet, and Alex Grossman in France (a mathematician, a geophysicist, and a theoretical physicist, respectively). With their common interest in time-frequency localization and multiresolution analysis, they built a framework and dubbed their creation ondelette (little wave), which became "wavelet" in English. The subject really caught on with the innovations of Ingrid Daubechies and Stephane Mallat, which had direct applicability to signal processing, and a veritable explosion of activity in wavelet theory and application ensued.

What are Wavelets Used For? Here, we describe three general fields of application in which wavelets have had a substantial impact, then we briefly explore the relationships these fields have with statistical analysis.

1. Signal processing Perhaps the most common application of wavelets (and certainly the impetus behind much of their development) is in signal processing. A signal, broadly defined, is a sequence of numerical measurements, typically obtained electronically. This could be weather readings, a radio broadcast, or measurements from a seismograph. In signal processing, the interest lies in analyzing and coding the signal, with the eventual aim of transmitting the encoded signal so that it can be reconstructed with only minimal loss upon receipt. Signals are typically contaminated by random noise, and an important part of signal processing is accounting for this noise. A particular emphasis is on denoising, i.e., extracting the "true" (pure) signal from the noisy version actually observed. This endeavor is precisely the goal in statistical function estimation as well-to "smooth" the noisy data points to obtain an estimate of the underlying function. wavelets have performed admirably in both of these fields. Signal processors now have new, fast tools at their disposal that are

xvi

WHY WAVELETS?

well-suited for denoising signals, not only those with smooth, well-behaved natures, but also those signals with abrupt jumps, sharp spikes, and other irregularities. These advantages of wavelets translate directly over to statistical data analysis. If signal processing is to be done in "real time," i.e., if the signals are treated as they are observed, it is important that fast algorithms are implemented. It doesn't matter how well a particular de-noising technique works if the algorithm is too complex to work in real time. One of the key advantages that wavelets have in signal processing is the associated fast algorithms-faster, even, than the fast Fourier transform.

2. Image analysis Image analysis is actually a special case of signal processing, one that deals with two-dimensional signals representing digital pictures. Again, typically, random noise is included with the observed image, so the primary goal is again denoising. In image processing, the denoising is done with a specific purpose in mind: to transform a noisy image into a "nice-looking" image. Though there might not be widespread agreement as to how to quantify the "niceness" of a reconstructed image, the general aim is to remove as much of the noise as possible, but not at the expense of fine-scale details. Similarly, in statistics, it is important to those seeking analysis of their data that estimated regression functions have a nice appearance (they should be smooth), but sometimes the most important feature of a data set is a sharp peak or abrupt jump. Wavelets help in maintaining real features while smoothing out spurious ones, so as not to "throw out the baby with the bathwater."

3. Data compression Electronic means of data storage are constantly improving. At the same time, with the continued gathering of extensive satellite and medical image data, for example, amounts of data requiring storage are increasing too, placing a constant strain on current storage facilities. The aim in data compression is to transform an enormous data set, saving only the most important elements of the transformed data, so that it can be reconstructed later with only a minimum of loss. As an example, Wickerhauser (1994) reports that the United States Federal Bureau of Investigation (FBI) has collected 30 million sets of fingerprints. For these to be digitally scanned and stored in an easily accessible form would require an enormous amount of space, as each digital fingerprint requires about 0.6 megabytes of storage. Wavelets have proven extremely useful in solving such problems, often requiring less than 30 kilobytes of storage space for an adequate representation of the original data, an impressive compression ratio of 20: 1. How does this relate to problems in statistics? To quote Manny Parzen, "Statistics is like art is like dynamite: The goal is compression." In multiple

Why Wavelets?

xvii

linear regression, for example, it is desired to choose the simplest model that represents the data adequately, to achieve a parsimonious representation. With wavelets, a large data set can often be summarized well with only a relatively small number of wavelet coefficients. To summarize, there are three main answers to the question "Why wavelets?":

1. good time-frequency localization, 2. fast algorithms, 3. simplicity of form.

This chapter has spent some time covering Answer 1 and how it is important in statistics. Answer 2 is perhaps more important in pure signal processing applications, but it is certainly valuable in statistical analysis as well. Some brief comments on Answer 3 are in order here. An entire set of wavelet functions is constructed by means of two simple operations on a single prototype function (referred to earlier as the "mother wavelet"): dilation and translation. The prototype function need never be computed when taking the wavelet transform of data. Just as the Fourier transform describes a function in terms of simple functions (sines and cosines), the wavelet transform describes a function in terms of simple wavelet component functions. The nature of this book is expository. Thus, it consists of an introduction to wavelets and descriptions of various applications in data analysis. For many of the statistical problems treated, more than one methodology is discussed. While some discussion of relative advantages and disadvantages of each competing method is in order, Ultimately, the specific application of interest must guide the data analyst to choose the method best suited for his/her situation. In statistics and data analysis, there is certainly room for differences of opinion as to which method is most appropriate for a given application, so the discussion of various methods in this book stops short of making specific recommendations on which method is "best," leaving this entirely to the reader to determine. With the basic introduction of wavelets and their applications in this text, readers will gain the necessary background to continue their study of other applications and more advanced wavelet methods. As increasingly more researchers become interested in wavelet methods, the class of problems to which wavelets have application is rapidly expanding. The References section at the end of this book lists several articles not covered in this book that provide further reading on wavelet methods and applications. There are many good introductory papers on wavelets. Rioul and Vetterli (1991) give a basic introduction focusing on the signal processing uses

xviii

WHY WAVELETS?

of wavelets. Graps (1995) describes wavelets for a general audience, giving some historical background and describing various applications. Jawerth and Sweldens (1994) give a broad overview of practical and mathematical aspects of wavelet analysis. Statistical issues pertaining to the application of wavelets are given in Bock (1992), Bock and Pliego (1992), and Vidakovic and Miiller (1994). There have been many books written on the subject of wavelets as well. Some good references are Daubechies (1992), Chui (1992), and Kaiser (1994) -these are all at a higher mathematical level than this book. The book by Strang and Nguyen (1996) provides an excellent introduction to wavelets from an engineering/signal processing point of view. Echoing the assertion of Graps (1995), most of the work in developing the mathematical foundations of wavelets has been completed. It remains for us to study their applications in various areas. We now embark upon an exploration of wavelet uses in statistics and data analysis.

CHAPTER

ONE

Wavelets: A Brief Introduction

This chapter gives an introductory treatment of the basic ideas concerning wavelets. The wavelet decomposition of functions is related to the analogous Fourier decomposition, and the wavelet representation is presented first in terms of its simplest paradigm, the Haar basis. This piecewise constant Haar system is used to describe the concepts of the multiresolution analysis, and these ideas are generalized to other types of wavelet bases. This treatment is meant to be merely an introduction to the relevant concepts of wavelet analysis. As such, this chapter provides most of the background for the rest of this book. It is important to stress that this book covers only the essential elements of wavelet analysis. Here, we assume knowledge of only elementary linear algebra and calculus, along with a basic understanding of statistical theory. More advanced topics will be introduced as they are encountered.

1.1

The Discrete Fourier Transform

Transformation of a function into its wavelet components has much in common with transforming a function into its Fourier components. Thus, an introduction to wavelets begins with a discussion of the usual discrete Fourier transform. This discussion is not by any means intended to be a complete treatment of Fourier analysis, but merely an overview of the subject to highlight the concepts that will be important in the development of wavelet analysis. While studying heat conduction near the beginning of the nineteenth century, the French mathematician and physicist Jean-Baptiste Fourier discovered that he could decompose any of a large class of functions into component functions constructed of only standard periodic trigonometric func-

2

THE DISCRETE FOURIER TRANSFORM

tions. Here, we will only consider functions defined on the interval [-'If, 'If]. (If a particular function of interest g is defined instead on a different finite

interval [a, b], then it can be transformed via f(x) = g(2'1fxj(b - a) - (a + b) 'If j(b - a)).) The sine and cosine functions are defined on all of IR and have period 2'If , so the Fourier decomposition can be thought of either as representing all such periodic functions, or as representing functions defined only on [-'If, 'If] by simply restricting attention to only this interval. Here, we will take the latter approach. The Fourier representation applies to square-integrable functions. Specifically, we say that a function f belongs to the square-integrable function space L2[a, b] if

Fourier's result states that any function f E L2[_'lf, 'If] can be expressed as an infinite sum of dilated cosine and sine functions:

f(x)

1

00

= 2ao + ~)aj cos(jx) + bj sin(jx)) ,

(1.1)

j=1

for an appropriately computed set of coefficients {ao, aI, b1 , ••• }. A word of caution is in order about the representation (1.1). The equality is only meant in the L2 sense, i.e.,

It is possible that f and its Fourier representation differ on a few points (and this is, in fact, the case at discontinuity points). Since this book is concerned primarily with analyzing functions in L2 space, this point will usually be neglected hereafter in similar representations. It is important to keep in mind, however, that such an expression does not imply pointwise convergence. The summation in (1.1) is up to infinity, but a function can be well-approx-

imated (in the L2 sense) by a finite sum with upper summation limit index J: J

SJ(x) =

~ao + ~(aj cos(jx) + bj sin(jx)).

(1.2)

Wavelets: A Brief Introduction

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