Table Of Content

223 Graduate Texts in Mathematics Editorial Board S. Axler F.W. Gehring K.A. Ribet Graduate Texts in Mathematics TAKEUTJIZARING. Introduction to 34 SPITZER. Principles ofRandom Walk. Axiomatic Set Theory. 2nd ed. 2nded. 2 0XTOBY. Measure and Category. 2nd ed. 35 Al.BxANDERIWBRMBR. Several Complex 3 ScHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed. 2nd ed. 36 Km.u!YINAMIOKA et al. Linear 4 Hn.roNISTAMMBACH. A Course in Topologica) Spaces. Homological Algebra. 2nd ed. 37 MoNK. Mathematical Logic. 5 MAc LANE. Categories for the Working 38 GRAUERTIFRITzsCHE. Several Complex Mathematician. 2nd ed. Variables. 6 Humms/PIPER. Projective Planes. 39 AilVESON. An Jnvitation to C*-Aigebras. 7 J.-P. SERRE. A Course in Arithmetic. 40 KEMENY/SNEU.IKNAPP. Denumerable 8 TAKEUTJIZARING. Axiomatic Set Theory. Markov Chains. 2nd ed. 9 HUMPHREYS. Jntroduction to Lie Algebras 41 APosTOL. Modular Functions and and Representation Theory. Dirichlet Series in Number Theory. 10 CoHEN. A Course in Simple Homotopy 2nded. Theory. 42 1.-P. SERRE. Linear Representations of 11 CoNWAY. Functions ofOne Complex Finite Groups. Variable 1. 2nd ed. 43 GILLMANIJERISON. Rings ofContinuous 12 BEALS. Advanced Mathematical Analysis. Functions. 13 ANDERSON!Fuu.ER. Rings and Categories 44 KENDIG. Elementary Algebraic Geometry. of Modules. 2nd ed. 45 LotM!. Probability Theory 1. 4th ed. 14 GoLUBITSKYIGUILLEMIN. Stable Mappings 46 LotM!. Probability Theory n. 4th ed. and Their Singularities. 47 MoiSE. Geometric Topology in 15 BERBERIAN. Lectures in Functional Dimensions 2 and 3. Analysis and Operator Theory. 48 SACHSIWu. General Relativity for 16 WINTER. The Structure ofFields. Mathemaficians. 17 RoSENBLATT. Random Processes. 2nd ed. 49 GRUENBERGIWEIR.. Linear Geometry. 18 HAI.Mos. Measure Theory. 2nded. 19 HAI.Mos. A Hilbert Space Prob1em Book. 50 EDWARDS. Fermat's Last Theorem. 2nded. 51 KLINGENBERG. A Course in Differential 20 HUSEMOLLER. Fibre Bundles. 3rd ed. Geometry. 21 HUMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry. 22 BARNES!MACK. An Algebraic Introduction 53 MANIN. A Course in Mathematical Logic. to Mathematica1 Logic. 54 GRAV ERIWA TKINS. Combinatorics with 23 GREUB. Linear Algebra. 4th ed. Emphasis on the Theory of Graphs. 24 HoLMES. Geometric Functional Analysis 55 BROwN!PEARCY. lntroduction to Operator and lts Applications. Theory 1: Elements of Functional Analysis. 25 HEwm/STROMBERG. Real and Abstract 56 MASSEY. Algebraic Topology: An Analysis. Introduction. 26 MANES. Algebraic Theories. 57 CRoWELLIFox. Introduction to Knot 27 KELLEY. General Topology. Theory. 28 :lARISKIISAMUEL. Commutative Algebra. 58 Kosmz. p-adic Numbers, p-adic Vol.l. Analysis, and Zeta-Functions. 2nd ed. 29 ZAR.IsKIISAMUEL. Commutative Algebra. 59 I...ANG. Cyclotomic Fields. Voi. II. 60 ARNOLD. Mathematical Methods in 30 JACOBSON. Lectures in Abstract Algebra 1. Classical Mechanics. 2nd ed. Basic Concepts. 61 WHITEHEAD. Elements of Homotopy 31 JACOBSON. Lectures in Abstract Algebra D. Theory. Linear Algebra. 62 KARGAPOLOV/MERLZJAKOV. Fundamentals 32 JACOBSON. Lectures in Abstract Algebra of the Theory of Groups. III. Theory ofFields and Galois Theory. 63 BOLLOBAS. Graph Theory. 33 HlRsCH. Differential Topology. (continued after index) Anders Vretblad Fourier Analysis and lts Applications ~Springer Anders Vretblad Department of Mathematics Uppsala University Box 480 SE-751 06 Uppsala Sweden [email protected] Editorial Board: S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Mathematics Department San Francisco State East Hall University of California, University University of Michigan Berkeley San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840 USA USA USA axler@ sfsu.edu [email protected] [email protected] .edu Mathematics Subject Classification (2000): 42-01 Library of Congress Cataloging-in-Publication Data Vretblad, Anders. Fourier analysis and its applications 1 Anders Vretblad. p. cm. Includes bibliographical references and index. ISBN 0-387-00836-5 (hc. : alk. paper) 1. Fourier analysis. I. Title. QA403.5. V74 2003 515'2433-dc21 2003044941 ISBN 0-387-00836-5 Printed on acid-free paper. © 2003 Springer-Verlag New York, Inc. Ali rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, com puter software, or by similar or dissimilar methodology now known or hereafter developed is for bidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 98765432 Corrected second printing, 2005 SPIN 10920442 springeronline.com To YNGVE DOMAR, my teacher, mentor, and friend Preface The classical theory of Fourier series and integrals, as well as Lap lace trans forms, is of great importance for physical and technical applications, and its mathematical beauty makes it an interesting study for pure mathemati cians as well. I have taught courses on these subjects for decades to civil engineering students, and also mathematics majors, and the present volume can be regarded as my collected experiences from this work. There is, of course, an unsurpassable book on Fourier analysis, the trea tise by Katznelson from 1970. That book is, however, aimed at mathemat ically very mature students and can hardly be used in engineering courses. On the other end of the scale, there are a number of more-or-less cookbook styled books, where the emphasis is almost entirely on applications. I have felt the need for an alternative in between these extremes: a text for the ambitious and interested student, who on the other hand does not aspire to become an expert in the field. There do exist a few texts that fulfill these requirements (see the literature list at the end of the book), but they do not include all the topics I like to cover in my courses, such as Laplace transforms and the simplest facts about distributions. The reader is assumed to have studied real calculus and linear algebra and to be familiar with complex numbers and uniform convergence. On the other hand, we do not require the Lebesgue integral. Of course, this somewhat restricts the scope of some of the results proved in the text, but the reader who does master Lebesgue integrals can probably extrapolate the theorems. Our ambition has been to prove as much as possible within these restrictions. viii Some knowledge of the simplest distributions, such as point masses and dipoles, is essential for applications. I have chosen to approach this mat ter in two separate ways: first, in an intuitive way that may be sufficient for engineering students, in star-marked sections of Chapter 2 and sub sequent chapters; secondly, in a more strict way, in Chapter 8, where at least the fundaments are given in a mathematically correct way. Only the one-dimensional case is treated. This is not intended to be more than the merest introduction, to whet the reader's appetite. Acknowledgements. In my work I have, of course, been inspired by exist ing literature. In particular, I want to mention a book by Arne Broman, lntroduction to Partial Differential Equations ... (Addison-Wesley, 1970), a compendium by Jan Petersson of the Chalmers Institute of Technology in Gothenburg, and also a compendium from the Royal Institute of Technol ogy in Stockholm, by Jockum Aniansson, Michael Benedicks, and Karim Daho. I am grateful to my colleagues and friends in Uppsala. First of all Professor Yngve Domar, who has been my teacher and mentor, and who introduced me to the field. The book is dedicateq to him. 1 am also partic ularly indebted to Gunnar Berg, Christer O. Kiselman, Anders Kăllstrom, Lars-Ăke Lindahl, and Lennart Salling. Bengt Carlsson has helped with ideas for the applications to control theory. The problems have been worked and re-worked by Jonas Bjermo and Daniel Domert. If any incorrect an swers still remain, the blame is mine. Finally, special thanks go to three former students at Uppsala University, Mikael Nilsson, Matthias Palmer, and Magnus Sandberg. They used an early version of the text and presented me with very constructive criticism. This actually prompted me to pursue my work on the text, and to translate it into English. Uppsala, Sweden Anders Vret blad January 2003 Contents Preface vii 1 Introduction 1 101 The classical partial differential equations 1 1.2 Well-posed problems 3 o o o o o o o o o 1.3 The one-dimensional wave equation 5 o 1.4 Fourier's method 9 o o o o o o o o o o o 2 Preparations 15 201 Complex exponentials 15 o o o o o o o o o o o o 202 Complex-valued functions of a real variable 17 203 Cesaro summation of series 20 o o 204 Positive summation kernels 22 o o 205 The Riemann-Lebesgue lemma 25 206 *Some simple distributions 27 20 7 *Computing with 8 32 o o o 3 Laplace and Z transforms 39 301 The Laplace transform 39 o o o o o o o o 302 Operations 42 o o o o o o o o o o o o o o 303 Applications to differential equations 47 3.4 Convolution 53 o o o o o o o o o o o o o o 305 *Laplace transforms of distributions 57 306 The Z transform 60 o o o o o o o o o o o x Contents 3. 7 Applications in control theory . 67 Summary of Chapter 3 . . . . . . . . 70 4 Fourier series 73 4.1 Definitions ................. . 73 4.2 Dirichlet's and Fejer's kernels; uniqueness 80 4.3 Differentiable functions . . . 84 4.4 Pointwise convergence . . . 86 4.5 Formulae for other periods . 90 4.6 Some worked examples . . . 91 4. 7 The Gibbs phenomenon . . 93 4.8 *Fourier series for distributions 96 Summary of Chapter 4 . . . . . . . . 100 5 L2 Theory 105 5.1 Linear spaces over the complex numbers 105 5.2 Orthogonal projections . . . . . 110 5.3 Some examples . . . . . . . . . 114 5.4 The Fourier system is complete 119 5.5 Legendre polynomials . . . . . 123 5.6 Other classical orthogonal polynomials 127 Summary of Chapter 5 . . . . . . . . . . . . 130 6 Separation of variables 137 6.1 The solution of Fourier's problem . 137 6.2 Variations on Fourier's theme . . . 139 6.3 The Dirichlet problem in the unit disk 148 6.4 Sturm-Liouville problems . . . . . . . 153 6.5 Some singular Sturm-Liouville problems 159 Summary of Chapter 6 . . . . . . . . . . . . . 160 7 Fourier transforms 165 7.1 Introduction ............ . 165 7.2 Definition of the Fourier transform 166 7.3 Properties . . . . . . . . . 168 7.4 The inversion theorem . . 171 7.5 The convolution theorem. 176 7.6 Plancherel's formula 180 7.7 Application 1 ...... . 182 7.8 Application 2 ...... . 185 7.9 Application 3: The sampling theorem. 187 7.10 *Connection with the Laplace transform 188 7.11 *Distributions and Fourier transforms 190 Summary of Chapter 7 . . . . . . . . . . . . . 192