Cambridge A First Course In Fourier Analysis

This page intentionally left blank

A First Course in Fourier Analysis

This unique book provides ameaningful resource for appliedmathematics through Fourieranalysis. It develops a unied theory of discrete and continuous (univariate) Fourier analy-sis, the fast Fourier transform, and a powerful elementary theory of generalized functions,including the use of weak limits. It then shows how these mathematical ideas can beused to expedite the study of sampling theory, PDEs, wavelets, probability, diffraction, etc.Unique features include a unied development of Fourier synthesis/analysis for functionson R, Tp, Z, and PN; an unusually complete development of the Fourier transform cal-culus (for nding Fourier transforms, Fourier series, and DFTs); memorable derivations ofthe FFT; a balanced treatment of generalized functions that fosters mathematical under-standing as well as practical working skills; a careful introduction to Shannons samplingtheorem and modern variations; a study of the wave equation, diffusion equation, anddiffraction equation by using the Fourier transform calculus, generalized functions, andweak limits; an exceptionally efcient development of Daubechies compactly supportedorthogonalwavelets; generalized probability density functionswith corresponding versionsof Bochners theorem and the central limit theorem; and a real-world application of Fourieranalysis to the study of musical tones. A valuable reference on Fourier analysis for a vari-ety of scientic professionals, including Mathematicians, Physicists, Chemists, Geologists,Electrical Engineers, Mechanical Engineers, and others.

David Kammler is a Professor and Distinguished Teacher in the Mathematics Department atSouthern Illinois University.

A First Course inFourier Analysis

David W. KammlerDepartment of MathematicsSouthern Illinois University at Carbondale

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So Paulo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-88340-5

ISBN-13 978-0-521-70979-8

ISBN-13 978-0-511-37689-4

D. W. Kammler 2007

2008

Information on this title: www.cambridge.org/9780521883405

This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

paperback

eBook (EBL)

hardback

Mathematics: Source andSubstance

Profound study of nature is the most fertile source of mathematicaldiscoveries.

Joseph Fourier, The Analytical Study of Heat, p. 7

Mathematics is the science of patterns. The mathematician seekspatterns in number, in space, in science, in computers, and in imagina-tion. Mathematical theories explain the relations among patterns; func-tions and maps, operators and morphisms bind one type of pattern toanother to yield lasting mathematical structures. Applications of mathe-matics use these patterns to explain and predict natural phenomena thatt the patterns. Patterns suggest other patterns, often yielding patternsof patterns. In this way mathematics follows its own logic, beginning withpatterns from science and completing the portrait by adding all patternsthat derive from initial ones.

Lynn A. Steen, The science of patterns, Science 240(1988), 616.

Contents

Preface xi

The Mathematical Core

Chapter 1 Fouriers representation for functions onR, Tp, Z, and PN 1

1.1 Synthesis and analysis equations 11.2 Examples of Fouriers representation 121.3 The Parseval identities and related results 231.4 The FourierPoisson cube 311.5 The validity of Fouriers representation 37

Further reading 59Exercises 61

Chapter 2 Convolution of functions on R, Tp, Z, and PN 89

2.1 Formal denitions of f g, f g 892.2 Computation of f g 912.3 Mathematical properties of the convolution product 1022.4 Examples of convolution and correlation 107


Chapter 3 The calculus for nding Fouriertransforms of functions on R 129

3.1 Using the denition to nd Fourier transforms 1293.2 Rules for nding Fourier transforms 1343.3 Selected applications of the Fourier transform calculus 147


vii

viii Contents

Chapter 4 The calculus for nding Fouriertransforms of functions on Tp, Z, and PN 173

4.1 Fourier series 1734.2 Selected applications of Fourier series 1904.3 Discrete Fourier transforms 1964.4 Selected applications of the DFT calculus 212


Chapter 5 Operator identities associated withFourier analysis 239

5.1 The concept of an operator identity 2395.2 Operators generated by powers of F 2435.3 Operators related to complex conjugation 2515.4 Fourier transforms of operators 2555.5 Rules for Hartley transforms 2635.6 Hilbert transforms 266


Chapter 6 The fast Fourier transform 291

6.1 Pre-FFT computation of the DFT 2916.2 Derivation of the FFT via DFT rules 2966.3 The bit reversal permutation 3036.4 Sparse matrix factorization of F when N = 2m 3106.5 Sparse matrix factorization of H when N = 2m 3236.6 Sparse matrix factorization of F when N = P1P2 Pm 3276.7 Kronecker product factorization of F 338


Chapter 7 Generalized functions on R 367

7.1 The concept of a generalized function 3677.2 Common generalized functions 3797.3 Manipulation of generalized functions 3897.4 Derivatives and simple dierential equations 4057.5 The Fourier transform calculus for generalized

functions 4137.6 Limits of generalized functions 4277.7 Periodic generalized functions 4407.8 Alternative denitions for generalized functions 450


Contents ix

Selected Applications

Chapter 8 Sampling 483

8.1 Sampling and interpolation 4838.2 Reconstruction of f from its samples 4878.3 Reconstruction of f from samples of a1 f , a2 f , . . . 4978.4 Approximation of almost bandlimited functions 505


Chapter 9 Partial dierential equations 523

9.1 Introduction 5239.2 The wave equation 5269.3 The diusion equation 5409.4 The diraction equation 5539.5 Fast computation of frames for movies 571


Chapter 10 Wavelets 593

10.1 The Haar wavelets 59310.2 Support-limited wavelets 60910.3 Analysis and synthesis with Daubechies wavelets 64010.4 Filter banks 655


Chapter 11 Musical tones 693

11.1 Basic concepts 69311.2 Spectrograms 70211.3 Additive synthesis of tones 70711.4 FM synthesis of tones 71111.5 Synthesis of tones from noise 71811.6 Music with mathematical structure 723


x Contents

Chapter 12 Probability 737

12.1 Probability density functions on R 73712.2 Some mathematical tools 74112.3 The characteristic function 74612.4 Random variables 75312.5 The central limit theorem 764


Appendices A-1

Appendix 1 The impact of Fourier analysis A-1Appendix 2 Functions and their Fourier transforms A-4Appendix 3 The Fourier transform calculus A-14Appendix 4 Operators and their Fourier transforms A-19Appendix 5 The WhittakerRobinson ow chart

for harmonic analysis A-23Appendix 6 FORTRAN code for a radix 2 FFT A-27Appendix 7 The standard normal probability distribution A-33Appendix 8 Frequencies of the piano keyboard A-37

Index I-1

Preface

To the Student

This book is about one big idea: You can synthesize a variety of complicated func-tions from pure sinusoids in much the same way that you produce a major chordby striking nearby C, E, G keys on a piano. A geometric version of this idea formsthe basis for the ancient Hipparchus-Ptolemy model of planetary motion (Almagest,2nd century see Fig. 1.2). It was Joseph Fourier (Analytical Theory of Heat, 1815),however, who developed modern methods for using trigonometric series and inte-grals as he studied the ow of heat in solids. Today, Fourier analysis is a highlyevolved branch of mathematics with an incomparable range of applications and withan impact that is second to none (see Appendix 1). If you are a student in one ofthe mathematical, physical, or engineering sciences, you will almost certainly ndit necessary to learn the elements of this subject. My goal in writing this book isto help you acquire a working knowledge of Fourier analysis early in your career.

If you have mastered the usual core courses in calculus and linear algebra, youhave the maturity to follow the presentation without undue diculty. A few of theproofs and more theoretical exercises require concepts (uniform continuity, uniformconvergence, . . . ) from an analysis or advanced calculus course. You may choose toskip over the dicult steps in such arguments and simply accept the stated results.The text has been designed so that you can do this without severely impactingyour ability to learn the important ideas in the subsequent chapters. In addition, Iwill use a potpourri of notions from undergraduate courses in dierential equations[solve y(x) + y(x) = 0, y(x) = xy(x), y(x) + 2y(x) = 0, . . . ], complex analysis(Eulers formula: ei = cos +i sin , arithmetic for complex numbers, . . . ), numbertheory (integer addition and multiplication modulo N , Euclids gcd algorithm, . . . ),probability (random variable, mean, variance, . . . ), physics (F = ma, conservationof energy, Huygens principle, . . . ), signals and systems (LTI systems, low-passlters, the Nyquist rate, . . . ), etc. You will have no trouble picking up these conceptsas they are introduced in the text and exercises.

If you wish, you can nd additional information about almost any topic inthis book by consulting the annotated references at the end of the correspondingchapter. You will often discover that I have abandoned a traditional presentation

xi

;

xii Preface

in favor of one that is in keeping with my goal of making these ideas accessibleto undergraduates. For example, the usual presentation of the Schwartz theoryof distributions assumes some familiarity with the Lebesgue integral and witha graduate-level functional analysis course. In contrast, my development of ,X, . . . in Chapter 7 uses only notions from elementary calculus. Once you masterthis theory, you can use generalized functions to study sampling, PDEs, wavelets,probability, diraction, . . . .

The exercises (541 of them) are my greatest gift to you! Read each chaptercarefully to acquire the basic concepts, and then solve as many problems as youcan. You may nd it benecial to organize an interdisciplinary study group, e.g.,mathematician + physicist + electrical engineer. Some of the exercises provideroutine drill: You must learn to nd convolution products, to use the FT calculus,to do routine computations with generalized functions, etc. Some supply historicalperspective: You can play Gauss and discover the FFT, analyze Michelson andStrattons analog supercomputer for summing Fourier series, etc. Some ask formathematical details: Give a sucient condition for . . . , given an example of . . . ,show that, . . . . Some involve your personal harmonic analyzers: Experimentallydetermine the bandwidth of your eye, describe what would you hear if you replacenotes with frequencies f1, f2, . . . by notes with frequencies C/f1, C/f2, . . . . Someprepare you for computer projects: Compute to 1000 digits, prepare a movie fora vibrating string, generate the sound le for Rissets endless glissando, etc. Somewill set you up to discover a pattern, formulate a conjecture, and prove a theorem.(Its quite a thrill when you get the hang of it!) I expect you to spend a lot of timeworking exercises, but I want to help you work eciently. Complicated results arebroken into simple steps so you can do (a), then (b), then (c), . . . until you reachthe goal. I frequently supply hints that will lead you to a productive line of inquiry.You will sharpen your problem-solving skills as you take this course.

Synopsis xiii

Synopsis

The chapters of the book are arranged as follows:

........................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................ ...........................................................................

....

....

....

....

....

....

....

...

....

....

......................................................................................................................

........................................................................................................................................................................................................................................ ...........................................................................

....

....

....

....

....

....

....

....

....

...

...................................................................................................................... ........................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................

....................................

....................................

....................................

....................................

....................................

...................................

...................................

.................................

.................................

............................. .......

......................

...............

1Fouriers

Representation

2Convolution

3FT Calculus

R

4FT CalculusTp,Z,PN

7GeneralizedFunctions

12Probability

5FT Operators

6The FFT

8Sampling

10Wavelets

11MusicalTones

9PDEs

The mathematical core is given in Chapters 17 and selected applications aredeveloped in Chapters 812.

We present the basic themes of Fourier analysis in the rst two chapters.Chapter 1 opens with Fouriers synthesis and analysis equations for functions on thereal line R, on the circle Tp, on the integers Z, and on the polygon PN . We discretize

xiv Preface

by sampling (obtaining functions on Z,PN from functions on R,Tp), we periodize bysumming translates (obtaining functions on Tp,PN from functions on R,Z), and weinformally derive the corresponding Poisson identities. We combine these mappingsto form the FourierPoisson cube, a structure that links the Fourier transforms,Fourier series, and discrete Fourier transforms students encounter in their under-graduate classes. We prove that these equations are valid when certain elementarysucient conditions are satised. We complete the presentation of basic themes bydescribing the convolution product of functions on R,Tp,Z, and PN in Chapter 2.

Chapters 3 and 4 are devoted to the development of computational skills. Weintroduce the Fourier transform calculus for functions on R by nding transformsof the box, (x), the truncated exponential, ex h(x), and the unit gaussian ex

2.

We present the rules (linearity, translation, dilation, convolution, inversion, . . . )and use them to obtain transforms for a large class of functions on R. Variousmethods are used to nd Fourier series. In addition to direct integration (withKroneckers rule), we present (and emphasize) Poissons formula, Eagles method,and the use of elementary Laurent series (from calculus). Corresponding rulesfacilitate the manipulation of the Fourier representations for functions on Tp and Z.An understanding of the Fourier transform calculus for functions on PN is essentialfor anyone who wishes to use the FFT. We establish a few well-known DFT pairsand develop the corresponding rules. We illustrate the power of this calculus byderiving the EulerMaclaurin sum formula from elementary numerical analysis andevaluating the Gauss sums from elementary number theory.

In Chapter 5 we use operators, i.e., function-to-function mappings, to organizethe multiplicity of specialized Fourier transform rules. We characterize the basicsymmetries of Fourier analysis and develop a deeper understanding of the Fouriertransform calculus. We also use the operator notation to facilitate a study of Sine,Cosine, Hartley, and Hilbert transforms.

The subject of Chapter 6 is the FFT (which Gilbert Strang calls the most impor-tant algorithm of the 20th century!). After describing the O(N2) scheme of Horner,we use the DFT calculus to produce an N -point DFT with only O(N log2 N) op-erations. We use an elementary zipper identity to obtain a sparse factorization ofthe DFT matrix and develop a corresponding algorithm (including the clever en-hancements of Bracewell and Buneman) for fast machine computation. We brieyintroduce some of the more specialized DFT factorizations that can be obtained byusing Kronecker products.

An elementary exposition of generalized functions (the tempered distributions ofSchwartz) is given in Chapter 7, the heart of the book. We introduce the Dirac [asthe second derivative of the ramp r(x) := max(x, 0)], the comb X; the reciprocal1/x, the Fresnel function eix

2, . . . and carefully extend the FT calculus rules to

this new setting. We introduce generalized (weak) limits so that we can work withinnite series, innite products, ordinary derivatives, partial derivatives, . . . .

Selected applications of Fourier analysis are given in the remaining chapters.(You can nd whole textbooks devoted to each of these topics.) Mathematical

To the Instructor xv

models based on Fourier synthesis, analysis done with generalized functions, andFFT computations are used to foster insight and understanding. You will experiencethe enormous leverage Fourier analysis can give as you study this material!

Sampling theory, the mathematical basis for digital signal processing, is the focusof Chapter 8. We present weak and strong versions of Shannons theorem togetherwith the clever generalization of Papoulis. Using these ideas (and characteristicsof the human ear) we develop the elements of computer music in Chapter 11. Weuse additive synthesis and Chownings FM synthesis to generate samples for musicaltones, and we use spectrograms to visualize the structure of the corresponding soundles.

Fourier analysis was invented to solve PDEs, the subject of Chapter 9. We for-mulate mathematical models for the motion of a vibrating string, for the diusionof heat (Fouriers work), and for Fresnel diraction. (The Schrodinger equationfrom quantum mechanics seems much less intimidating when interpreted within thecontext of elementary optics!) With minimal eort, we solve these PDEs, establishsuitable conservation laws, and examine representative solutions. (The cover illus-tration was produced by using the FFT to generate slices for the diraction patternthat results when two gaussian laser beams interfere.)

Chapter 10 is devoted to the study of wavelets, a relatively new branch of math-ematics. We introduce the basic ideas using the piecewise constant functions asso-ciated with the Haar wavelets. We then use the theory of generalized functions todevelop the compactly supported orthogonal wavelets created by I. Daubechies in1988. Fourier analysis plays an essential role in the study of corresponding lterbanks that are used to process audio and image les.

We present the elements of probability theory in Chapter 12 using generalizeddensities, e.g., f(x) := (1/2)[(x + 1) + (x 1)] serves as the probability densityfor a coin toss. We use Fourier analysis to nd moments, convolution products,characteristic functions, and to establish the uncertainty relation (for suitably reg-ular probability densities on R). We then use the theory of generalized functions toprove the central limit theorem, the foundation for modern statistics!

To the Instructor

This book is the result of my eorts to create a modern elementary introduction toFourier analysis for students from mathematics, science, and engineering. There ismore than enough material for a tight one-semester survey or for a leisurely two-semester course that allocates more time to the applications. You can adjust thelevel and the emphasis of the course to your students by the topics you cover andby your assignment of homework exercises. You can use Chapters 14, 7, and 9 toupdate a lackluster boundary value problems course. You can use Chapters 14, 7,8, and 10 to give a serious introduction to sampling theory and wavelets. You can

xvi Preface

use selected portions of Chapters 24, 6, 8, and 11 (with composition exercises!)for a fascinating elementary introduction to the mathematics of computer-generatedmusic. You can use the book for an undergraduate capstone course that emphasizesgroup learning of the interdisciplinary topics and mastering of some of the moredicult exercises. Finally, you can use Chapters 712 to give a graduate-levelintroduction to generalized functions for scientists and engineers.

This book is not a traditional mathematics text. You will nd a minimal amountof jargon and note the absence of a logically complete theorem-proof presentation ofelementary harmonic analysis. Basic computational skills are developed for solvingreal problems, not just for drill. There is a strong emphasis on the visualization ofequations, mappings, theorems, . . . and on the interpretation of mathematical ideaswithin the context of some application. In general, the presentation is informal,but there are careful proofs for theorems that have strategic importance, and thereare a number of exercises that lead students to develop the implications of ideasintroduced in the text.

Be sure to cover one or more of the applications chapters. Students enjoy learningabout the essential role Fourier analysis plays in modern mathematics, science, andengineering. You will nd that it is much easier to develop and to maintain themarket for a course that emphasizes these applications.

When I teach this material I devote 24 lectures to the mathematical core (deletingportions of Chapters 1, 5, and 6) and 18 lectures to the applications (deletingportions of Chapters 10, 11, and 12). I also spend 34 hours per week conductinginformal problem sessions, giving individualized instruction, etc. I lecture fromtransparencies and use a PC (with FOURIER) for visualization and sonication.This is helpful for the material in Chapters 2, 5, 6, and 12 and essential for thematerial in Chapters 9, 10, and 11. I use a laser with apertures on 35 mm slidesto show a variety of diraction patterns when I introduce the topic of diractionin Chapter 9. This course is a great place to demonstrate the synergistic rolesof experimentation, mathematical modeling, and computer simulation in modernscience and engineering.

I have one word of caution. As you teach this material you will face the constanttemptation to prove too much too soon. My informal use of ?= cries out for theprecise statement and proof of some relevant sucient condition. (In most casesthere is a corresponding exercise, with hints, for the student who would really liketo see the details.) For every hour that you spend presenting 19th-century advancedcalculus arguments, however, you will have one less hour for explaining the 20th-century mathematics of generalized functions, sampling theory, wavelets, . . . . Youmust decide which of these alternatives will best serve your students.

Acknowledgments

I wish to thank Southern Illinois University at Carbondale for providing a nurtur-ing environment during the evolution of ideas that led to this book. Sabbatical

Acknowledgments xvii

leaves and teaching fellowships were essential during the early phases of the work.National Science Foundation funding (NSF-USE 89503, NSF-USE 9054179) for fourfaculty short courses at the Touch of Nature center at SIUC during 19891992, andNECUSE funding for a faculty short course at Bates College in 1995 enabled meto share preliminary versions of these ideas with faculty peers. I have protedenormously from their thoughtful comments and suggestions, particularly thoseof Davis Cope, Bo Green, Carruth McGehee, Dale Mugler, Mark Pinsky, DavidSnider, Patrick Sullivan, Henry Warchall, and Jo Ward. I deeply appreciate theNational Science Foundation course development grants (NSF-USE 9156064, NSF-CCLI 127048) that provided support for the creation of many of the exercise setsas well as for equipment and programming services that were used to develop thesecond half of the book.

I wish to thank my editor, David Tranah, and Cambridge University Press forproducing a corrected 2nd edition of this book. I appreciate the meticulous atten-tion to detail shown by Viji Muralidhar, production manager at Newgen ImagingSystems, as the book was prespared for publication. I am indebted to George Lobelland Prentice-Hall for their initial investment in this work. I am delighted to ac-knowledge the encouraging reviews of the book published by Simon Godsill (LondonTimes Higher Education Supplement, 24 Nov 2000), David Snider (IEEE Spectrum,Dec 2000), Carruth McGehee (MAA Monthly, Oct 2001), and Chris Heil (SIAM Re-view, Dec 2001). I appreciate the critical role that Pat Van Fleet, David Eubanks,Xinmin Li, Wenbing Zhang, Je McCreight and Fawaz Hjouj played as graduatestudents in helping me to learn the details associated with various applications ofFourier analysis. I am particularly indebted to David Eubanks for the innumerablehours he invested in the development of the software package FOURIER that I usewhen I teach this material. I want to acknowledge my debt to Rebecca Parkin-son for creating the charming sketch of Joseph Fourier that appears in Fig. 3.4.My heartfelt thanks go to Linda Gibson and to Charles Gibson for preparing theoriginal TEX les for the book (and for superhuman patience with a myriad of revi-sions!). Finally, I express my deep appreciation to my wife, Ruth, for her love andencouragement throughout this project.

I hope that you enjoy this approach for learning Fourier analysis. If you havecorrections, ideas for new exercises, suggestions for improving the presentation, etc.,I would love to hear from you!

David W. KammlerMathematics DepartmentSouthern Illinois University at CarbondaleCarbondale, Illinois [email protected]

1Fouriers representation forfunctions on R, Tp, Z, and PN

1.1 Synthesis and analysis equations

Introduction

In mathematics we often try to synthesize a rather arbitrary function f using asuitable linear combination of certain elementary basis functions. For example, thepower functions 1, x, x2, . . . serve as such basis functions when we synthesize f usingthe power series representation

f(x) = a0 + a1x + a2x2 + . (1)

The coecient ak that species the amount of the basis function xk needed in therecipe (1) for constructing f is given by the well-known Maclaurin formula

ak =f (k)(0)

k!, k = 0, 1, 2, . . .

from elementary calculus. Since the equations for a0, a1, a2, . . . can be used onlyin cases where f, f , f , . . . are dened at x = 0, we see that not all functionscan be synthesized in this way. The class of analytic functions that do have suchpower series representations is a large and important one, however, and like Newton[who with justiable pride referred to the representation (1) as my method], youhave undoubtedly made use of such power series to evaluate functions, to constructantiderivatives, to compute denite integrals, to solve dierential equations, to jus-tify discretization procedures of numerical analysis, etc.

1

2 Fouriers representation for functions

Fouriers representation (developed a century and a half after Newtons) uses asbasis functions the complex exponentials

e2isx := cos(2sx) + i sin(2sx), (2)

where s is a real frequency parameter that serves to specify the rate of oscillation,and i2 = 1. When we graph this complex exponential, i.e., when we graph

u := Re e2isx = cos(2sx)

v := Im e2isx = sin(2sx)

as functions of the real variable x in x, u, v-space, we obtain a helix (a Slinky!)that has the spacing 1/|s| between the coils. Projections of this helix on the planesv = 0, u = 0, x = 0 give the sinusoids u = cos(2sx), v = sin(2sx), and the circleu2 + v2 = 1, as shown in Fig. 1.1.

Figure 1.1. The helix u = cos(2sx), v = sin(2sx) inx, u, v-space together with projections in the x, u, the x, v,and the u, v planes.

Synthesis and analysis equations 3

Functions on R

Fourier discovered that any suitably regular complex-valued function f dened onthe real line R can be synthesized by using the integral representation

f(x) = s=

F (s)e2isx ds, < x < . (3)

Here F is also a complex-valued function dened on R, and we think of F (s)dsas being the amount of the exponential e2isx with frequency s that must be usedin the recipe (3) for f . At this point we are purposefully vague as to the exacthypotheses that must be imposed on f to guarantee the existence of such a Fourierrepresentation. Roughly speaking, the Fourier representation (3) is possible in allcases where f does not uctuate too wildly and where the tails of f at are nottoo large. It is certainly not obvious that such functions can be represented in theform (3) [nor is it obvious that sinx, cosx, ex, and many other functions can berepresented using the power series (1)]. At this point we are merely announcing thatthis is, in fact, the case, and we encourage you to become familiar with equation (3)along with analogous equations that will be introduced in the next few paragraphs.Later on we will establish the validity of (3) after giving meaning to the intentionallyvague term suitably regular.

Fourier found that the auxiliary function F from the representation (3) can beconstructed by using the integral

F (s) = x=

f(x)e2isx dx, < s < . (4)

We refer to (3) as the synthesis equation and to (4) as the analysis equation for f .The function F is said to be the Fourier transform of f . We cannot help but noticethe symmetry between (3) and (4), i.e., we can interchange f, F provided that wealso interchange +i and i. Other less symmetric analysis-synthesis equations aresometimes used for Fouriers representation, see Ex. 1.4, but we prefer to use (3)(4)in this text. We will often display the graphs of f, F side by side, as illustrated inFig. 1.2. Our sketch corresponds to the case where both f and F are real valued.In general, it is necessary to display the four graphs of Re f , Im f , ReF , and ImF.You will nd such displays in Chapter 3, where we develop an ecient calculus forevaluating improper integrals having the form (3) or (4).

Figure 1.2. The graph of a function f on R and its Fourier transform F on R.


Functions on Tp

We say that a function f dened on R is p-periodic, p > 0, when

f(x + p) = f(x), < x < .

Fourier (like Euler, Lagrange, and D. Bernoulli before him) discovered that a suit-ably regular p-periodic complex-valued function on R can be synthesized by usingthe p-periodic complex exponentials from (2). We will routinely identify any p-periodic function on R with a corresponding function dened on the circle Tp hav-ing the circumference p as illustrated in Fig. 1.3. [To visualize the process, think ofwrapping the graph of f(x) versus x around a right circular cylinder just like thepaper label is wrapped around a can of soup!] Of course, separate graphs for Re fand Im f must be given in cases where f is complex valued.

Figure 1.3. Identication of a p-periodic function f on R witha corresponding function on the circle Tp having the circumfer-ence p.

The complex exponential e2isx will be p-periodic in the argument x, i.e.,

e2is(x+p) = e2isx, < x < ,

whene2isp = 1,

i.e., whens = k/p for some k = 0, 1, 2, . . . .

In this way we see that the p-periodic exponentials from (2) are given by

e2ikx/p, k = 0, 1, 2, . . . ,

as shown in Fig. 1.4.


Figure 1.4. Real and imaginary parts of the complex exponen-tial e8ix/p as functions on R and as functions on Tp.

Fouriers representation

f(x) =

k=F [k]e2ikx/p, < x < , (5)

for a p-periodic function f uses all of these complex exponentials. In this caseF is a complex-valued function dened on the integers Z (from the German wordZahlen, for integers). We use brackets [ ] rather than parentheses ( ) to enclose theindependent variable k in order to remind ourselves that this argument is discrete.We think of F [k] as being the amount of the exponential e2ikx/p that we must usein the recipe (5) for f . We refer to (5) as the Fourier series for f and we say thatF [k] is the kth Fourier coecient for f . You may be familiar with the alternativerepresentation

f(x) =a02

+

k=1

{ak cos(2kx/p) + bk sin(2kx/p)}

for a Fourier series. You can use Eulers identity (2) to see that this representationis equivalent to (5), see Ex. 1.16. From time to time we will work with such cos, sin


series, e.g., this form may be preferable when f is real or when f is known to haveeven or odd symmetry. For general purposes, however, we will use the compactcomplex form (5).

Fourier found that the coecients F [k] for the representation (5) can be con-structed for a given function f by using the integrals

F [k] =1p

px=0

f(x)e2ikx/p dx, k = 0,1,2, . . . . (6)

[Before discovering the simple formula (6), Fourier made use of clumsy, mathemat-ically suspect arguments based on power series to nd these coecients.] We referto (5) as the synthesis equation and to (6) as the analysis equation for the p-periodicfunction f , and we say that F is the Fourier transform of f within this context. Weuse small circles on line segments, i.e., lollipops, when we graph F (a function onZ), and we often display the graphs of f, F side by side as illustrated in Fig. 1.5. Ofcourse, we must provide separate graphs for Re f , Im f , ReF , ImF in cases wheref, F are not real valued. You will nd such displays in Chapter 4, where we developa calculus for evaluating integrals having the form (6).

Figure 1.5. The graph of a function f on Tp and its Fouriertransform F on Z.

Functions on Z

There is a Fourier representation for any suitably regular complex-valued functionf that is dened on the set of integers, Z. As expected, we synthesize f from thecomplex exponential functions e2isn on Z, with s being a real parameter. Now forany real s and any integer m we nd

e2i(s+m)n = e2isn, n = 0,1,2, . . .(i.e., the exponentials e2isn, e2i(s1)n, e2i(s2)n, . . . are indistinguishable whenn is constrained to take integer values). This being the case, we will synthesize fusing

e2isn, 0 s < 1


or equivalently, usinge2isn/p, 0 s < p,

where p is some xed positive number. Figure 1.6 illustrates what happens whenwe attempt to use some s > p. The high-frequency sinusoid takes on the identity oralias of some corresponding low-frequency sinusoid. It is easy to see that e2isn/poscillates slowly when s is near 0 or when s is near p. The choice s = p/2 gives themost rapid oscillation with the complex exponential

e2i(p/2)n/p = (1)n

having the smallest possible period, 2.

Figure 1.6. The identical samples of e2ix/16 and e2i17x/16 atx = 0,1,2, . . . .

Fouriers synthesis equation,

f [n] = ps=0

F (s)e2isn/p ds, (7)

for a suitably regular function f on Z, uses all of these complex exponentials on Z,and the corresponding analysis equation is given by

F (s) =1p

n=

f [n]e2isn/p. (8)

We say that F is the Fourier transform of f and observe that this function isp-periodic in s, i.e., that F is a complex-valued function on the circle Tp. Figure 1.7illustrates such an f, F pair.


Figure 1.7. The graph of a function f on Z and its Fouriertransform F on Tp.

We have chosen to include the parameter p > 0 for the representation (7) (insteadof working with the special case p = 1) in order to emphasize the duality that existsbetween (5)(6) and (7)(8). Indeed, if we replace

i, x, k, f, F

in (5)(6) byi, s, n, pF, f,

respectively, we obtain (7)(8). Thus every Fourier representation of the form(5)(6) corresponds to a Fourier representation of the form (7)(8), and vice versa.

Functions on PN

Let N be a positive integer, and let PN consist of N uniformly spaced points on thecircle TN as illustrated in Fig. 1.8. We will call this discrete circle a polygon evenin the degenerate cases where N = 1, 2.

Figure 1.8. The polygon P5.

The simplest Fourier representation [found by Gauss in the course of his studyof interpolation by trigonometric polynomials a few years before Fourier discoveredeither (3)(4) or (5)(6)] occurs when f is a complex-valued N -periodic functiondened on Z. We will routinely identify such an N -periodic f with a correspondingfunction that is dened on PN as illustrated in Fig. 1.9. Of course, we must provideseparate graphs for Re f , Im f when f is complex valued. Since f is completelyspecied by the N function values f [n], n = 0, 1, . . . , N 1, we will sometimes ndthat it is convenient to use a complex N -vector

f = (f [0], f [1], . . . , f [N 1])


Figure 1.9. Identication of an N -periodic discrete function onZ with a corresponding function on the polygon PN .

to represent this function. This is particularly useful when we wish to process fnumerically. You will observe that we always use n = 0, 1, . . . , N 1 (not n =1, 2, . . . , N) to index the components of f .

The complex exponential e2isn (with s being a xed real parameter) will beN -periodic in the integer argument n, i.e.,

e2is(n+N) = e2isn for all n = 0, 1, 2, . . .

whene2isN = 1,

i.e., when s = k/N for some integer k. On the other hand, when m is an integer wend

e2ikn/N = e2i(k+mN)n/N for all n = 0, 1, 2, . . . ,so the parameters

s =k

N, s =

k NN

, s =k 2N

N, . . .

all give the same function. Thus we are left with precisely N distinct discreteN -periodic complex exponentials

e2ikn/N , k = 0, 1, . . . , N 1.

The complex exponentials with k = 1 or k = N 1 make one complete oscillationon PN , those with k = 2 or k = N 2 make two complete oscillations, etc., asillustrated in Fig. 1.10. The most rapid oscillation occurs when N is even andk = N/2 with the corresponding complex exponential

e2i(N/2)n/N = (1)n

having the smallest possible period, 2.


Figure 1.10. Complex exponentials e2ikn/31 on P31.

Fouriers synthesis equation takes the form

f [n] =N1k=0

F [k]e2ikn/N , n = 0, 1, 2, . . . (9)

within this setting. Again we regard F [k] as the amount of the discrete exponentiale2ikn/N that must be used in the recipe for f , we refer to (9) as the discreteFourier series for f , and we say that F [k] is the kth Fourier coecient for f . Thecorresponding analysis equation

F [k] =1N

N1n=0

f [n]e2ikn/N , k = 0, 1, . . . , N 1 (10)


enables us to nd the coecients F [0], F [1], . . . , F [N 1] for the representation(9) from the known function values f [0], f [1], . . . , f [N 1]. We refer to F as thediscrete Fourier transform (DFT) or more simply as the Fourier transform of fwithin this context. The formula (10) gives an N -periodic discrete function on Zwhen we allow k to take all integer values, so we will say that F is a function onPN . Again, we plot graphs of f, F side by side, as illustrated in Fig. 1.11. Youwill nd such displays in Chapter 4, where we develop a calculus for evaluating thenite sums (9), (10). Later on, in Chapter 6, you will learn an ecient way to dosuch calculations on a computer.

Figure 1.11. The graph of a function f on P31 and its Fouriertransform F on P31.

Summary

The following observations will help you remember Fouriers synthesis and analysisequations:

Functions on R Functions on Tp

(3) f(x) = s=

F (s)e2isxds (5) f(x) =

k=F [k]e2ikx/p

(4) F (s) = x=

f(x)e2isxdx (6) F [k] =1p

px=0

f(x)e2ikx/pdx

Functions on Z Functions on PN

(7) f [n] = ps=0

F (s)e2isn/pds (9) f [n] =N1k=0

F [k]e2ikn/N

(8) F (s) =1p

n=

f [n]e2isn/p (10) F [k] =1N

N1n=0

f [n]e2ikn/N .

The Fourier transform F has the real argument s when f is aperiodic and theinteger argument k when f is periodic. The function f has the real argument xwhen F is aperiodic and the integer argument n when F is periodic.


The argument of the exponentials that appear in the synthesis-analysis equationsis the product of 2i, the argument s or k of F , the argument x or n of f , andthe reciprocal 1/p or 1/N of the period if either f or F is periodic.

The synthesis equation uses the +i exponential and all values of F to form f .The analysis equation uses the i exponential and all values of f to form F.

The reciprocal 1/p or 1/N of the period serves as a scale factor on the analysisequation in cases where either f or F is periodic. No such factor is used on thesynthesis equation.

During the opening scene of an opera you catch glimpses of the main characters,but you have not yet learned the subtle personality traits or relationships thatwill unfold during the rest of the performance. In much the same way, you havebeen briey introduced to the eight remarkable identities (3)(10) that will appearthroughout this course. (You can verify this by skimming through the text!) At thispoint, it would be very benecial for you to spend a bit of time getting acquaintedwith these identities. Begin with a function f from Exs. 1.1, 1.81.10, 1.13, 1.14,evaluate the sum or integral from the analysis equation to nd F , and then evaluatethe sum or integral from the synthesis equation to establish the validity of Fouriersrepresentation for this f . (It is hard to nd examples where both of these sums,integrals can be found by using the tools from calculus!) See if you can determinehow certain symmetries possessed by f are made manifest in F by doing Exs. 1.2,1.11, 1.15. Explore alternative ways for writing the synthesis-analysis equations asgiven in Exs. 1.3, 1.4, 1.12, 1.16. And if you are interested in associating somephysical meaning with the synthesis and analysis equations, then do try Ex. 1.17!

1.2 Examples of Fouriers representation

Introduction

What can you do with Fouriers representation? In this section we will brieydescribe six diverse settings for these ideas that will help you learn to recognize thepatterns (3)(10). Other applications will be developed with much more detail inChapters 812. (You may want to read the rst few pages of some of these chaptersat this time!)

The HipparchusPtolemy model of planetary motion

One of the most dicult problems faced by the ancient Greek astronomers wasthat of predicting the position of the planets. A remarkably successful model ofplanetary motion that is described in Ptolemys Almagest leads to an interestinggeometric interpretation for truncated Fourier series. Using modern notation we

Examples of Fouriers representation 13

writez1(t) = a1e2it/T1 , < t <

witha1 = |a1|ei1 , 0 1 < 2

to describe the uniform circular motion of a planet P around the Earth E at theorigin. Here |a1| is the radius of the orbit, T1 is the period, and the phase parameter1 serves to specify the location of the planet at time t = 0. Such a one-circle modelcannot account for the occasional retrograde motion of the outer planets Mars,Jupiter, and Saturn. We build a more sophisticated two-circle model by writing

z2(t) = z1(t) + a2e2it/T2

witha2 = |a2|ei2 , 0 2 < 2.

The planet P now undergoes uniform circular motion about a point that under-goes uniform circular motion around the Earth E at the origin, see Fig. 1.12. Thistwo-circle model can produce the observed retrograde motion (try a computer sim-ulation using the data from Ex. 1.18!), but it cannot t the motion of the planetsto observational accuracy.

Figure 1.12. The addition of uniform circular motions.

Proceeding in this way we obtain a geometric interpretation of the motion de-scribed by the exponential sum

zn(t) = a1e2it/T1 + a2e2it/T2 + + ane2it/Tn

using a xed circle (called the deferent) and n 1 moving circles (called epicycles).Such a motion is periodic when T1, T2, . . . , Tn are integral multiples of some T > 0,


in which case the sum is a Fourier series with nitely many terms. Hipparchus andPtolemy used a shifted four-circle construction of this type (with the Earth nearbut not at the origin) to t the motion of each planet. These models were used forpredicting the positions of the ve planets of antiquity until Kepler and Newtondiscovered the laws of planetary motion some 1300 years later.

Gauss and the orbits of the asteroids

On the rst day of the 19th century the asteroid Ceres was discovered, and inrapid succession the asteroids Pallas, Vesta, and Juno were also found. Gaussbecame interested in the problem of determining the orbits of such planetoids fromobservational data. In 1802, Baron von Zach published the 12 data points for theorbit of the asteroid Pallas that are plotted in Fig. 1.13. Gauss decided to interpolatethis data by using a 360-periodic trigonometric polynomial

y(x) =11

k=0

ck e2ikx/360

with the 12 coecients c0, c1, . . . , c11 being chosen to force the graph of y to passthrough the 12 known points (n 30, yn), n = 0, 1, . . . , 11, i.e., so as to make

yn =11

k=0

ck e2ikn/12, n = 0, 1, . . . , 11.

Figure 1.13. Declination of the asteriod Pallas as a function ofright ascension as published by Baron von Zach. (Declinationand right ascension are measures of latitude and longitude onthe celestial sphere.)


We recognize this as the synthesis equation (9) (with N = 12, F [k] = ck), anduse the corresponding analysis equation (10) to obtain the coecients

ck =112

11n=0

yn e2ikn/12, k = 0, 1, . . . , 11.

Of course, it is one thing to write down such a formula and quite another to obtain anumerical value for each of the cks. (Remember that Gauss did all of the arithmeticby hand.) You will nd Baron von Zachs data in Ex. 1.19. Perhaps as you analyzethis data (with a computer!) you will share in Gausss discovery of a very cleverway to expedite such calculations.

Fourier and the ow of heat

Shortly after the above work of Gauss was completed, Fourier invented the rep-resentations (5)(6) and (3)(4) (i.e., Fourier series and Fourier integrals) to usefor solving problems involving the ow of heat in solids. He rst showed that thetemperature u(x, t) at time t 0 and coordinate x along a thin insulated rod ofuniform cross section is a solution of the partial dierential equation

u

t(x, t) = a2

2u

x2(x, t)

with the thermal diusivity parameter a2 depending on the material of which the rodis made. (You will nd an elementary derivation in Section 9.3.) Fourier observedthat the function

e2isx e42a2s2tsatises the partial dierential equation for every choice of the real parameter s. Heconceived the idea of combining such elementary solutions to produce a temperaturefunction u(x, t) that agrees with some prescribed initial temperature when t = 0.

For the temperature in a rod (that extends from x = to x = +) Fourierwrote

u(x, t) = s=

A(s)e2isxe42a2s2t ds

with the intention of choosing the amplitude function A(s), < s < , to makehis formula to agree with the known initial temperature u(x, 0) at time t = 0, i.e.,to make

u(x, 0) = s=

A(s)e2isx ds.

We recognize this identity as the synthesis equation (3) for the function u(x, 0) anduse the corresponding analysis equation (4) to write

A(s) = x=

u(x, 0)e2isx dx, < s < ,


thereby expressing A in terms of the initial temperature. In this way Fourier solvedthe heat ow problem for a doubly innite rod. You can work out the details for arod with an initial hot spot by solving Ex. 1.20.

For the temperature in a ring of circumference p > 0, Fourier used the p-periodicsolutions

e2ikx/p e42a2(k/p)2t, k = 0,1,2, . . .of the diusion equations (with s = k/p) to write

u(x, t) =

k=ck e

2ikx/p e42a2(k/p)2t

with the intention of choosing the coecients ck, k = 0,1,2, . . . to make

u(x, 0) =

k=ck e

2ikx/p.

We recognize this as the synthesis equation (5) for the initial temperature u(x, 0)and use the corresponding analysis equation (6) to express the coecients

ck =1p

px=0

u(x, 0)e2ikx/p dx, k = 0,1,2, . . .

in terms of the known initial temperature. In this way Fourier solved the heat owproblem for a ring.

Today, such procedures are used to solve a number of partial dierential equationsthat arise in science and engineering, and we will develop these ideas in Chapter 9.It is somewhat astonishing, however, to realize that Fourier chose periodic functionsto study the ow of heat, a physical phenomenon that is as intrinsically aperiodicas any that we can imagine!

Fouriers representation and LTI systems

Function-to-function mappings are commonly studied in many areas of science andengineering. Within the context of engineering we focus attention on the device thateects the input-to-output transformation, and we represent such a system using adiagram of the sort shown in Fig. 1.14. In mathematics, such function-to-functionmappings are called operators, and we use the notation

fo = Afi

orA : fi fo by fo(t) = (Afi)(t)

(where A is the name of the operator) to convey the same idea.


fiInput

SystemAfo

Output

Figure 1.14. Schematic representation of a system A.

In practice we often deal with systems that are homogeneous and additive, i.e.,

A(cf) = c(Af)A(f + g) = (Af) + (Ag)

when f, g are arbitrary inputs and c is an arbitrary scalar. Such systems are said tobe linear. Many common systems also have the property of translation invariance.We say that a system is translation invariant if the output

go = Agi

of an arbitrary -translate

gi(t) := fi(t + ), < t < ,of an arbitrary input function fi is the corresponding -translate

go(t) = fo(t + ), < t < of the output

fo = Afi

to fi, i.e., when we translate fi by the system responds by shifting fo by , < < . Systems that are both linear and translation invariant are said tobe LTI.

A variety of signal processing devices can be modeled by using LTI systems. Forexample, the speaker for an audio system maps an electrical input signal from anamplier to an acoustical output signal, with time being the independent variable.A well-designed speaker is more-or-less linear. If we simultaneously input signalsfrom two ampliers, the speaker responds with the sum of the corresponding out-puts, and if we scale the input signal, e.g., by adjusting the volume control, theacoustical response is scaled in a corresponding manner (provided that we do notexceed the power limitations of the speaker!) Of course, when we play a familiar CDor tape on dierent occasions, i.e., when we time shift the input signal, we expectto hear an acoustical response that is time shifted in exactly the same fashion (pro-vided that the time shift amounts to a few hours or days and not to a few millionyears!)

A major reason for the importance of Fourier analysis in electrical engineering isthat every complex exponential

es(t) := e2ist, < t <


(with s being a xed real parameter) is an eigenfunction of every LTI system. Wesummarize this by saying, An LTI system responds sinusoidally when it is shakensinusoidally. The proof is based on the familiar multiplicative property

es(t + ) = es() es(t)

of the complex exponential. After applying the LTI operator A to both sides ofthis equation, we use the translation invariance to simplify the left side, we use thelinearity to simplify the right side, and thereby write

(Aes)(t + ) = es() (Aes)(t), < t < , < < .

We now set t = 0 to obtain the eigenfunction relation

(Aes)() = (s) es(), < <

with the system function

(s) := (Aes)(0), < s <

being the corresponding eigenvalue.If we know the system function (s), < s < , we can nd the system

response to any suitably regular input function fi. Indeed, using Fouriers repre-sentation (3) we write

fi(t) = s=

Fi(s)e2ist ds

and approximate the integral of this synthesis equation with a Riemann sum of theform

fi(t) N

k=1

Fi(sk)e2iskt sk.

Since the linear operator A maps

e2ist to (s)e2ist

for every choice of the frequency parameter s, it must map the Riemann sum

Nk=1

Fi(sk)e2isktsk toN

k=1

Fi(sk)(sk)e2isktsk,

with the sum on the right being an approximation to the integral s=

Fi(s)(s)e2ist ds.


We conclude that A maps

fi(t) =

Fi(s)e2istds to fo(t) =

Fi(s)(s)e2ist ds

(provided that the system possesses a continuity property that enables us to justifythe limiting process involved in passing from an approximating Riemann sum tothe corresponding integral). In this way we see that the Fourier transform of theoutput is obtained by multiplying the Fourier transform of the input by the LTIsystem function .

The above discussion deals with systems that map functions on R to functionson R. Analogous considerations can be used for LTI systems that map functions onTp, Z, PN to functions on Tp, Z, PN , respectively, see Ex. 1.21.

Schoenbergs derivation of the TartagliaCardan formulas

The discrete Fourier representation of (9)(10) can be used to nd formulas for theroots of polynomials of degree 2, 3, 4 (see I. Schoenberg, pp. 7981). To illustratethe idea, we will derive the familiar quadratic formula for the roots x0, x1 of thequadratic polynomial

x2 + bx + c = (x x0)(x x1)as functions of the coecients b, c. In view of the synthesis equation (9) we canwrite

x0 = X0 + X1, x1 = X0 X1(where we take N = 2 and use x0, x1, X0, X1 instead of the more cumbersome x[0],x[1], X[0], X[1]). It follows that

x2 + bx + c ={x (X0 + X1)

}{x (X0 X1)

}= x2 2X0x + (X20 X21 ),

and upon equating coecients of like powers of x we nd

b = 2X0, c = X20 X21 .

We solve for X0, X1 in turn and write

X0 = 12b, X1 = 12 (b2 4c)1/2.

Knowing X0, X1 we use the synthesis equation to obtain the familiar expressions

x0 = 12{b + (b2 4c)1/2}, x1 = 12{b (b2 4c)1/2}.


The same procedure enables us to derive the TartagliaCardan formulas for theroots x0, x1, x2 of the cubic polynomial

x3 + bx2 + cx + d = (x x0)(x x1)(x x2)in terms of the coecients b, c, d. We dene

:= e2i/3 =1 + 3i

2so that we can use the compact form of the synthesis equations:

x0 = X0 + X1 + X2, x1 = X0 + X1 + 2X2, x2 = X0 + 2X1 + X2

to express x0, x1, x2 in terms of the discrete Fourier transform X0, X1, X2. After abit of nasty algebra (see Ex. 1.22) we nd

x3 + bx2 + cx + d = (x X0)3 3X1X2(x X0) X31 X32so that

X0 = b3 , X1X2 =b2 3c

9, X31 + X

32 =

27d + 9bc 2b327

.

From the last pair of equations we see that Y = X31 , X32 are the roots of the quadratic

polynomial

(Y X31 )(Y X32 ) = Y 2 (X31 + X32 )Y + (X1X2)3

= Y 2 +(27d 9bc + 2b3

27

)Y +

(b2 3c

9

)3,

i.e.,

X1 =

(27d 9bc + 2b3

54

)+

[(27d 9bc + 2b3

54

)2(

b2 3c9

)3]1/2

1/3

,

X2 =

(27d 9bc + 2b3

54

)[(

27d 9bc + 2b354

)2(

b2 3c9

)3]1/2

1/3

.

Knowing X0, X1, X2 we use the synthesis equation to write

x0 = X0 + X1 + X2,

x1 = X0 X1 + X22 +i3(X1 X2)

2,

x2 = X0 X1 + X22 i3(X1 X2)

2.

The roots of a quartic polynomial can be found in a similar manner (but it takesa lot of very nasty algebra to do the job!).


Fourier transforms and spectroscopy

We can produce an exponentially damped complex exponential

y0(t) :={et e2is0t if t > 00 if t < 0

by subjecting a damped harmonic oscillator (e.g., a mass on a spring with damping)to a suitable initial excitation. Here > 0 and < s0 < . Graphs of y0 andthe Fourier transform

Y0(s) = 0

e2ist et e2is0tdt =1

+ 2i(s s0)are shown in Fig. 1.15. The function Y0, which is called a Lorenzian, is concentratednear s = s0 with an approximate width /2. (You can learn more about suchfunctions by doing Ex. 3.34 a little later in the course.)

Figure 1.15. The function y0(t) = ete2is0t and its Fouriertransform Y0(s).

When we subject molecules to a burst of electromagnetic radiation (radio fre-quency, microwave, infrared, . . . ) we induce various damped oscillations. Theresulting transient has the form

y(t) =

k

Akekt e2iskt if t > 0

0 if t < 0

with parameters k > 0, < sk < that depend on the arrangement of theatoms that form the molecules. We can observe these parameters when we graphthe Fourier transform

Y (s) =k

Akk + 2i(s sk) .

Within this context Y is said to be a spectrum.


For example, when a sample of the amino acid arginine

O C C C C C CN NH+2

O NH+3 H H H H NH2

H H H H

is placed in a strong magnetic eld and subjected to a 500-MHz pulse, the individualprotons precess. The resulting free induction decay voltage, y(t), and correspondingspectrum, Y (s), are shown in Fig. 1.16. (You can see the individual Lorenzians!)Richard Earnst won the 1991 Nobel prize in chemistry for developing this idea intoa powerful tool for determining the structure of organic molecules.

Figure 1.16. FT-NMR analysis of arginine.

The Parseval identities and related results 23

1.3 The Parseval identities and related results

The Parseval identities

Let f, g be suitably regular functions on R with Fourier transforms F,G, respec-tively. Using the synthesis equation for g and the analysis equation for f (and usinga bar to denote the complex conjugate), we formally write

x=

f(x)g(x) dx = x=

f(x)

{ s=

G(s)e2isx ds

}dx

?= s=

x=

f(x)e2isxG(s) dx ds

= s=

F (s)G(s) ds,

assuming that we can somehow justify the exchange in the order of the integrationprocesses in the step marked with the question mark (e.g., by imposing restrictivehypotheses on f, g and using a suitable Fubini theorem from advanced calculus).We refer to the resulting equation

x=

f(x)g(x) dx = s=

F (s)G(s) ds (11)

as the Parseval identity for functions on R. Analogous arguments lead to the cor-responding Parseval identities

px=0

f(x)g(x) dx = p

k=F [k]G[k] (12)

n=

f [n]g[n] = p ps=0

F (s)G(s) ds (13)

N1n=0

f [n]g[n] = NN1k=0

F [k]G[k] (14)

for functions on Tp, Z, PN , respectively. The period p or N appears as a factoron the transform side of these equations. An exchange of innite summation andintegration processes is involved in this heuristic derivation of (12)(13). In contrast,only nite sums are used in the derivation of (14) from the synthesis and analysisequations (9)(10) that we will establish in a subsequent discussion. You will ndalternative forms for the Parseval identities (11)(14) in Ex. 1.24.


The Plancherel identities

When we set g = f in (11)(14) we obtain the equations

x=

|f(x)|2 dx = s=

|F (s)|2 ds, (15)

px=0

|f(x)|2 dx = p

k=|F [k]|2, (16)

n=

|f [n]|2 = p p0

|F (s)|2 ds, (17)

N1n=0

|f [n]|2 = NN1k=0

|F [k]|2 (18)

that link the aggregate squared size (or energy) of a function f on R, Tp, Z,PN , respectively, to that of its Fourier transform F . We will refer to (15)(18)as the Plancherel identities (although the names of Bessel, Lyapunov, Parseval, andRayleigh are also properly associated with these equations).

As we have noted, (15)(18) can be obtained from (11)(14) simply by settingg = f . The corresponding identities are really equivalent, however, since we can ob-tain a Parseval identity from the corresponding (seemingly less general) Plancherelidentity by using the polarization identities

fg = 14{|f + g|2 + i|f + ig|2 + i2|f + i2g|2 + i3|f + i3g|2},

F G = 14{|F + G|2 + i|F + iG|2 + i2|F + i2G|2 + i3|F + i3G|2}

together with the linearity of the Fourier transform process, see Ex. 1.25.

Orthogonality relations for the periodic complex exponentials

It is a simple matter to verify the orthogonality relations

px=0

e2ikx/p e2ix/p dx ={p if k = ,0 otherwise,

k, = 0,1,2, . . . (19)


for the p-periodic complex exponentials on R. The corresponding discrete orthogo-nality relations

N1n=0

e2ikn/Ne2in/N

={N if k = , N, 2N, . . ., k, = 0,1,2, . . .0 otherwise,

(20)

can be proved by using the formula

1 + z + z2 + + zN1 ={N if z = 1(zN 1)/(z 1) otherwise

for the sum of a geometric progression with

z := e2i(k)/N .

We easily verify that

z = 1 if k = 0,N,2N, . . .

whilezN = 1 for all k, = 0,1,2, . . .

and thereby complete the argument. An alternative geometric proof of (20) is theobject of Ex. 1.26. Real versions of (19)(20) are developed in Ex. 1.27.

The orthogonality relations (19), (20) are the special cases of the Parseval iden-tities (12), (14) that result when the discrete functions F,G vanish at all but oneof the points of Z,PN , respectively, where the value 1 is taken.

Bessels inequality

Let f be a function on Tp and let

n(x) :=n

k=ncke

2ikx/p (21)

be any p-periodic trigonometric polynomial of degree n or less with complex coe-cients ck, k = 0,1,2, . . . ,n. By using the analysis equation (6) for the Fourier


coecients of f and the orthogonality relations (19), we nd px=0

|f(x) n(x)|2 dx

= px=0

{f(x)

nk=n

ck e2ikx/p

}{f(x)

n=n

c e2ix/p

}dx

= px=0

|f(x)|2 dx n

=nc

px=0

f(x)e2ix/p dx

n

k=nck

px=0

f(x)e2ikx/p dx

+n

k=n

n=n

ckc

px=0

e2ikx/p e2ix/p dx

= px=0

|f(x)|2 dx pn

=nc F [] p

nk=n

ck F [k] + pn

k=nckck

= px=0

|f(x)|2 dx pn

k=n|F [k]|2 + p

nk=n

|F [k] ck|2 (22)

when all of the integrals exist and are nite, e.g., as is certainly the case when f isbounded and continuous at all but nitely many points of Tp.

If we specialize (22) by taking ck = F [k] for k = 0,1,2, . . . ,n, the rightmostsum vanishes and we nd p

x=0|f(x)|2 dx p

nk=n

|F [k]|2 = p0

f(x) n

k=nF [k]e2ikx/p

2

dx 0

for every choice of n = 1, 2, . . . . In this way we prove Bessels inequality, px=0

|f(x)|2dx p

k=|F [k]|2, (23)

a one-sided version of (16).

The Weierstrass approximation theorem

Let f be a continuous function on Tp. We will show that we can uniformly approx-imate f as closely as we please with a p-periodic trigonometric polynomial (21).More specically, we will construct trigonometric polynomials 1, 2, . . . such that

limn max0xp

|f(x) n(x)| = 0.


This result is known as the Weierstrass approximation theorem. We need this resultto establish the validity of (5)(6). The proof will use a few ideas from intermediateanalysis. (You may wish to jump to Section 1.4and come back later to sort out thedetails.)

For each n = 1, 2, . . . we dene the de la ValleePoussin power kernel

n(x) := p1 4n(2nn

)1cos2n(x/p), (24)

shown in Fig. 1.17. We will show that this nonnegative function has a unit areaconcentrated at the origin of Tp. By using the Euler identity for cos and the binomialformula, we write

n(x) = p1(2nn

)1{eix/p + eix/p

}2n

= p1(2nn

)1{(2n0

)e2inx/p +

(2n1

)e2i(n1)x/p

+(2n2

)e2i(n2)x/p +

+(2nn

)1 + +

(2n2n

)e2inx/p

}. (25)

Figure 1.17. The de la ValleePoussin power kernel (24) forn = 101, 102, 103.


Moreover, after noting that

n(0) = p14n(2nn

)1=

4nn!n!p(2n)!

=2np

2n 22n 1

2n 42n 3

23

0 is given. We rst choose 0 < < p/2 so small that

|f(x) f(u)| < /2 when |x u| < .With thus chosen, we use the tail hypothesis of ( 26) and select n so large that

max|x| p/2

n(x) 0 is some uniform bound for |f(x)|. By using the unit area hypothesisof ( 26) with the p-periodicity of n we see that

f(x) = pu=0

f(x)n(x u) du

[since f(x) is a constant with respect to the u-integration]. It follows that

f(x) n(x) = pu=0

{f(x) f(u)}n(x u) du

= x+p/2u=xp/2

{f(x) f(u)}n(x u) du.

In conjunction with the positivity hypothesis of (26) and our choices for , n thisleads to the uniform bound

|f(x) n(x)|

|xu||f(x) f(u)|n(x u) du

+ |xu| p/2

|f(x) f(u)|n(x u) du

max|xu|

|f(x) f(u)| pu=0

n(x u) du+ max

|xu| p/2|f(x) f(u)| n(x u) p

2

1 + 2M 4Mp

p = ,

thus completing the proof.You can use variations of this argument to study the pointwise convergence of

Fourier series, see Exs. 1.31, 1.32.There is a second (mean square) form of the Weierstrass approximation theorem

that can be used when f is bounded on Tp and continuous at all points of Tp butx1, x2, . . . , xm where jumps occur. In this case we can show that

limn

p0

|f(x) n(x)|2 dx = 0


for suitably chosen trigonometric polynomials 1, 2, . . . . We will form n as before,noting that |n(x)| M when M is a uniform bound for f . We let J() be the por-tion of Tp that remains after we remove small open intervals I1(), I2(), . . . , Im()of length centered at x1, x2, . . . , xm. We can then write

p0

|f(x) n(x)|2 dx m

=1

I()

|f(x) n(x)|2 dx +J()

|f(x) n(x)|2 dx

m (2M)2 + p maxxJ()

|f(x) n(x)|2.

Given > 0 we can make

m (2M)2 < 2

by choosing a suciently small . Since f is continuous on J(), the above argumentshows that

limn maxxJ()

|f(x) n(x)| = 0,

so we will havep max

xJ()|f(x) n(x)|2 < 2

for all suciently large n.

A proof of Plancherels identity for functions on Tp

Let f be a piecewise continuous function on Tp. We drop the nonnegative rightmostsum from (22) to obtain the inequality p

x=0|f(x)|2 dx p

nk=n

|F [k]|2 px=0

|f(x) n(x)|2 dx

whenever n is any p-periodic trigonometric polynomial (21) of degree n or less. Wehave shown that the right-hand side vanishes in the limit as n when we usethe construction (27), (29) of de la ValleePoussin to produce 1, 2, . . . , and in thisway we see that p

x=0|f(x)|2dx p

k=

|F [k]|2 0.

In conjunction with the Bessel inequality (23), this proves the Plancherel identity(16) for all piecewise continuous functions f on Tp. A proof of the Plancherelidentity (15) for suitably restricted functions f on R is given in Ex. 1.40.

Two essentially dierent piecewise continuous functions f, g on Tp cannot havethe same Fourier coecients. Indeed, if F [k] = G[k] for all k = 0,1,2, . . . , then

The FourierPoisson cube 31

we can use Plancherels identity to write

p0

|f(x) g(x)|2dx = p

k=|F [k] G[k]|2 = 0.

It follows that f(x) = g(x) at all points x where f and g are continuous.

1.4 The FourierPoisson cube

Introduction

Classical applications of Fourier analysis use the integral (3) or the innite series (5).Digital computers can be programmed to evaluate the nite sums (9)(10) withgreat eciency, see Chapter 6. We are now going to derive some identities thatconnect these seemingly unrelated forms of Fourier analysis. This will make itpossible for us to use discrete Fourier analysis to prepare computer simulations forvibrating strings, diusing heat, diracting light, etc. (as described in Section 9.5).

The synthesisanalysis equations (3)(4), (5)(6), (7)(8), (9)(10) establish bidi-rectional mappings f F , g G, , that link suitably regular func-tions f, g, , dened on R, Tp, Z, PN and their corresponding Fourier transformsF,G,,. We will formally establish certain connections between these four kindsof univariate Fourier analysis. In so doing, we introduce eight unidirectional map-pings f g, f , g , , F G, F , G , that serve tolink the unconnected adjacent corners of the incomplete cube of Fig. 1.19. In thisway we begin the process of unifying the various Fourier representations, and weprepare some very useful computational tools.

Figure 1.19. Functions from the four Fourier transform pairs (3)(4),(5)(6), (7)(8), and (9)(10) arranged on the corners of a cube.


Discretization by h -sampling

Given a function f on the continuum R and a spacing parameter h > 0, we canconstruct a corresponding discrete function on Z by dening

[n] := f(nh), n = 0,1,2, . . . .

We say that is constructed from f by h-sampling. The same process can be usedto construct a discrete function on PN from a function g on the continuum Tp, i.e.,to construct an N -periodic function on Z from a p-periodic function on R, but inthis case we must take h := p/N (so that N steps of size h will equal the period p).With this in mind we dene

[n] := g(np

N

), n = 0,1,2, . . . .

These discretization mappings f and g are illustrated in Fig. 1.20. Thediscrete functions , provide good representations for f, g in cases where f, g donot vary appreciably over any interval of length h.

Figure 1.20. Construction of functions , on Z,PN from func-tions f, g on R,Tp by h-sampling.

Periodization by p -summation

Let f be a function on R and assume that f(x) rapidly approaches 0 as x .We can sum the translates

. . ., f(x + 2p), f(x + p), f(x), f(x p), f(x 2p), . . .


to produce the p-periodic function

g(x) :=

m=f(x mp), < x <

when p > 0. We say that the function g on Tp is produced from f by p-summation.Analogously, when is a function on Z and [n] rapidly approaches 0 as n we can construct a function on PN , N = 1, 2, . . . by writing

[n] :=

m=[n mN ], n = 0,1,2, . . . .

These periodization mappings f g and are illustrated in Fig. 1.21. Theperiodic functions g, provide good representations for f, when the graphs of f, are concentrated in intervals of length p,N , respectively.

The Poisson relations

Let be a function on Z. We will assume that is absolutely summable, i.e.,

m=

|[m]| < ,

(to ensure that the above sum for [n] is convergent) and use the analysis equation(10) to obtain the discrete Fourier transform

[k] =1N

N1n=0

[n]e2ikn/N

=1N

N1n=0

m=

[n mN ]e2ikn/N .

Now since e2ikn/N is N -periodic in n and since every integer has a uniquerepresentation

= n mN with n = 0, 1, . . . , N 1 and m = 0,1,2, . . . ,we can write

[k] =1N

m=

N1n=0

[n mN ]e2ik(nmN)/N

=1N

=

[]e2ik/N .


Figure 1.21. Construction of functions g, on Tp,PN from func-tions f, on R,Z by p-summation, N -summation, respectively.

We now use the analysis equation (8) (with p replaced by q to avoid confusion at alater point in the presentation) to obtain

[k] =q

N 1q

=

[]e2i(kq/N)/q

=q

N(

kq

N

), k = 0,1,2, . . . .


If we construct from by N -summation, then we can obtain from by q/N -sampling and q/N -scaling. (The Fourier transform of is assumed to be afunction on Tq, q > 0.)

Analogously, when f is a suitably regular function on R we can nd the Fouriercoecients of the p-periodic function

g(x) :=

m=f(x mp)

by writing

G[k] =1p

px=0

g(x)e2ikx/p dx

=1p

px=0

m=

f(x mp)e2ikx/p dx

?=1p

m=

p0

f(x mp)e2ik(xmp)/p dx

=1p

=

f()e2ik/p d

=1pF

(k

p

), k = 0,1,2, . . . .

Of course, we must impose a mild regularity condition on f to ensure that thefunctions g,G are well dened and to ensure that the exchange of the summationand integration processes is permissible. In this way we see that if g is formed fromf by p-summation, then G is formed from F by 1/p-sampling and 1/p-scaling.

We have used the analysis equations (4) and (6), (8) and (10) to obtain theFourier transform pairs

g(x) :=

m=f(x mp), G[k] = 1

pF

(k

p

), (29)

[n] :=

m=[n mN ], [k] = q

N(

kq

N

), (30)

when f, are suitably regular functions on R,Z with Fourier transforms F, onR,Tq, respectively. Analogous arguments [using the synthesis equations (3) and(7), (5) and (9)] can be used to obtain the Fourier transform pairs

[n] := f(np

N

), (s) =

m=

F

(s mN

p

), (31)

[n] := g(np

N

), [k] =

m=

G[k mN ], (32)


when f, g are suitably regular functions on R,Tp with Fourier transforms F,G onR,Z, respectively, see Ex. 1.34. We will refer to (29)(32) as the Poisson rela-tions. You will observe the dual roles played by sampling and summation in theseequations.

The FourierPoisson cube

We use the Poisson relations together with the analysis and synthesis equations ofFourier (as arranged in Fig. 1.19) to produce the FourierPoisson cube of Fig. 1.22.Suitably regular functions f, g, , that are dened on R,Tp,Z,PN lie at the cornersof the left face of this cube, and the corresponding Fourier transforms F,G,,dened on R,Z,TN/p,PN , respectively, lie on the corners of the right face. Ofnecessity we must work with both p-periodic and q-periodic functions with q = N/pin this diagram (and this is why we introduced the parameter q in the previoussection). The synthesisanalysis equations (3)(10) allow us to pass back and forthfrom function to transform. The process of h-sampling and p-summation provide uswith one-way mappings that connect adjacent corners of the left (function) face ofthe cube, and Poissons formulas (29)(32) induce corresponding one-way mappingsthat connect adjacent corners of the right (transform) face of the cube.

Figure 1.22. The FourierPoisson cube is a commuting diagramformed from the 8 mappings of (3)(10) and the 8 mappings of(29)(32).

The validity of Fouriers representation 37

You will observe that it is possible to move from the f corner to the corneralong the alternative routes f g or f . We use the Poisson relationsto verify that these mappings produce the same function :

[n] = g(np

N

)=

m=

f(np

N mp

),

[n] =

m=[n mN ] =

m=

f([n mN ] p

N

).

You can use similar arguments to verify that any two paths joining one corner ofthe cube to another (in a way that is consistent with the arrows) correspond to thesame composite mapping. We summarize this by saying that the FourierPoissoncube is a commuting diagram.

The FourierPoisson cube is a helpful way to visualize the connections between(3)(10) and (29)(32). You will learn to work with all of these mappings as thecourse progresses. Practical methods for nding Fourier transforms of functions onR, i.e., for using the mappings f F , will be developed in Chapter 3. The Fourierseries mappings g G, and the DFT mappings will be studied inChapter 4. You will learn to use the equivalence of f g G and f F Gto nd many Fourier series with minimal eort! The fast Fourier transform (FFT),an ecient algorithm for eecting the mappings on a computer, will be thefocus of Chapter 6. You will even learn to invert the one-way discretization mapsf , g (when F,G are suitably localized) as you study the sampling theoremin Chapter 8. At this point, however, you will nd it most helpful to work throughExs. 1.35, 1.36 so that you will see how Poissons relations can be used to analyzethe error associated with certain discrete approximations to the integrals (6), (4)for Fourier transforms on Tp,R, respectively.

1.5 The validity of Fouriers representation

Introduction

In this section, we will establish the validity of Fouriers representation for suit-ably regular functions on PN ,Z,Tp,R and some of the arguments use ideas fromintermediate analysis. Focus on the ow of the argument as you read the proof forthe rst time, skipping over the steps that you do not understand. You can comeback and sort out the troublesome details after you have studied the more concretematerial in Chapters 24.

We will continue to use the letter pairs f, F , g,G, ,, and , (instead of thegeneric f, F of (3)(4), (5)(6), (7)(8), and (9)(10)) to help you follow the courseof the argument as we move around the FourierPoisson cube, establishing in turnthe links , , g G, and nally, f F.


Functions on PN

Let be any function on PN , i.e., let the complex numbers [0], [1], . . . , [N1]be given. Using the analysis equation (10) we dene

[k] :=1N

N1m=0

[m]e2ikm/N , k = 0, 1, . . . , N 1.

By using this expression together with the orthogonality relations (20) we nd

N1k=0

[k]e2ikn/N =N1k=0

{1N

N1m=0

[m]e2ikm/N}e2ikn/N

=N1m=0

[m]{

1N

N1k=0

e2ikn/N e2ikm/N}

= [n], n = 0, 1, . . . , N 1,i.e., the synthesis equation (9) holds. Thus we see that Fouriers representation canbe used for any function on PN , so the bottom front link from the FourierPoissoncube of Fig. 1.22 is secure.

Absolutely summable functions on Z

Let be an absolutely summable function on Z, i.e., [n] 0 as n sorapidly that

n=

|[n]| < .

This hypothesis of absolute summability ensures that the Fourier transform

(s) :=1q

n=

[n]e2isn/q

is well dened, with the series converging absolutely and uniformly on R to thecontinuous q-periodic function . Moreover, the same hypothesis guarantees thatthe N -periodic discrete function

[n] :=

m=[n mN ], n = 0,1,2, . . .

is well dened by N -summation with the corresponding discrete Fourier transformbeing given by the Poisson relation

[k] =q

N(

kq

N

), k = 0,1,2, . . .


of (30). We use these expressions for , in the synthesis equation

[n] =N1k=0

[k]e2ink/N , n = 0,1,2, . . .

(which we have just established) to obtain the discrete Poisson sum formula

m=

[n mN ] = qN

N1k=0

(

kq

N

)e2i(kq/N)(n/q), n = 0,1,2, . . . . (33)

As N , the translates [n mN ], m = 1,2, . . . from the sum on theleft of (33) move o to , while the Riemann sums on the right converge toa corresponding integral. Thus in the limit as N (33) yields the Fouriersynthesis equation

[n] = qs=0

(s)e2isn/q ds, n = 0,1,2, . . . .

In this way we prove that Fouriers representation (7)(8) is valid for any absolutelysummable function on Z. The four links at the bottom of the FourierPoisson cubeare secure when is such a function.

Continuous piecewise smooth functions on Tp

Let g be a continuous piecewise smooth function on Tp, i.e., g is continuous on Tpand g is dened and continuous at all but a nite number of points of Tp wherenite jump discontinuities can occur. The graph of g on Tp is thus formed fromnitely many smooth curves joined end-to-end with corners being allowed at thepoints of connection, e.g., as illustrated in Fig. 1.3. The Fourier coecients

G[k] :=1p

p0

g(x)e2ikx/p dx, k = 0,1,2, . . . ,

G1[k] :=1p

p0

g(x)e2ikx/pdx, k = 0,1,2, . . .

of g, g are then well dened. Since g(0+) = g(0) = g(p), we can use an integra-tion by parts argument to verify that

G1[k] = (2ik/p)G[k], k = 0,1,2, . . . .

We use this identity with the real inequality

|ab| 12 (a2 + b2)


and Bessels inequality (23) (for g) to see that

k=

|G[k]| = |G[0]| + p2

k =0

1kG1[k]

|G[0]| + p4

k =0

{1k2

+ |G1[k]|2}

|G[0]| + p2

k=1

1k2

+14

p0

|g(x)|2 dx

< .

Thus the function[k] := p G[k], k = 0,1,2, . . .

is absolutely summable on Z. We have shown that any such function has the Fourierrepresentation

[k] = px=0

(x)e2ikx/p dx, k = 0,1,2, . . . ,

where

(x) :=1p

k=

[k]e2ikx/p, < x < .

After expressing in terms of G this synthesisanalysis pair takes the form

G[k] =1p

px=0

(x)e2ikx/p dx, k = 0,1,2, . . .

(x) =

k=G[k]e2ikx/p, < x < .

In this way, we see that the original function g and the auxiliary function have thesame Fourier coecients. We apply the Plancherel identity (16) to the continuousp-periodic function g and thereby conclude that g = . In this way we establishthe desired synthesis equation

g(x) =

k=G[k]e2ikx/p, < x < ,

and prove that Fouriers representation (5)(6) is valid for any continuous piecewisesmooth function on Tp.


Since the Fourier coecients are absolutely summable, the sequence of partialsums

sn(x) :=n

k=nG[k]e2ikx/p, n = 0, 1, 2, . . .

of the Fourier series converges absolutely and uniformly on Tp to g. In particular,all but nitely many of the trigonometric polynomials s0, s1, s2, . . . have graphs thatlie within an arbitrarily small -tube

{(x, z): x Tp, z C, |g(x) z| <

}, > 0

drawn about the graph of g on Tp, see Fig. 1.23.

Figure 1.23. Any real -tube drawn about the graph of the con-tinuous piecewise smooth function f from Fig. 1.3 contains thegraphs of all but nitely many of the partial sums s0, s1, s2, . . .of the corresponding Fourier series.

The sawtooth singularity function on T1

In this section we study the convergence of the Fourier series of the 1-periodicsawtooth function

w0(x) :={0 if x = 012 x if 0 < x < 1

(34)

that is continuously dierentiable at all points of T1 except the origin, where aunit jump occurs, see Fig. 1.24. Using integration by parts we compute the Fouriercoecients

W0[k] := 10

(12

x)

e2ikxdx

= 12ik

{(12

x)

e2ikx1

x=0+ 10

e2ikxdx}

=1

2ik, k = 1,2, . . .


Figure 1.24. Graphs of the sawtooth singularity function (34)as a 1-periodic function on R and as a function on T1.

with

W0[0] = 10

( 12 x

)dx = 0.

We will show that the slightly modied Fourier representation

w0(x) = limL

Lk=L

W0[k]e2ikx =

k=1

sin(2kx)k

(35)

(with the limits at taken symmetrically) is valid at each point x. Since (35)holds trivially when x = 0, we need only give a proof for points 0 < x < 1.

We construct a continuous piecewise smooth 1-periodic function

w1(x) := 112 + xu=0

w0(u) du, < x < ,

having the derivative w0(x) (at points x = 0,1,2, . . .), noting that

w1(x) = 112 +x

2 x

2

2when 0 x 1.

The constant 1/12 has been chosen to make

W1[0] := 10

w1(x) dx = 0,

and an integration by parts argument can be used to verify that

W1[k] =1

(2ik)2, k = 1,2, . . . ,

see (4.19)(4.24) (i.e. equations (19)(24) from Chapter 4). We have already shownthat such a function w1 has the Fourier representation

w1(x) =

k=W1[k]e2ikx =

k=1

cos(2kx)k(2k)

, < x <


with this Fourier series converging absolutely and uniformly. This being the case,we can establish (35), i.e., we can justify the term-by-term dierentiation of theFourier series for w1, by showing that the series (35) converges uniformly on

Date post:	08-Dec-2016
Category:	Documents
Upload:	dangbao
View:	219 times
Download:	1 times

Cambridge A First Course In Fourier Analysis

Documents