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Fourier Analysis and Related Topics J. Korevaar Fourier Analysis and Related Topics J. Korevaar Preface For many years, the author taught a one-year course called “Mathe- matical Methods”. It was intended for beginning graduate students in the physical sciences and engineering, as well as for mathematics students with an interest in applications. The aim was to provide mathematical tools used in applications, and a certain theoretical background that would make other parts of mathematical analysis accessible to the student of physical science. The course was taken by a large number of students at the University of Wisconsin (Madison), the University of California San Diego (La Jolla), and finally, the University of Amsterdam. At one time the author planned to turn his elaborate lecture notes into a multi-volume book, but only one vol- ume appeared [68]. The material in the present book represents a selection from the lecture notes, with emphasis on Fourier theory. Starting with the classical theory for well-behaved functions, and passing through L1 and L2 theory, it culminates in distributional theory, with applications to bounday value problems. At the International Congress of Mathematicians (Cambridge, Mass) in 1950, many people became interested in the Generalized Functions or “Dis- tributions” of field medallist Laurent Schwartz; cf. [110]. Right after the congress, Michael Golomb, Merrill Shanks and the author organized a year- long seminar at Purdue University to study Schwartz’s work. The seminar led the author to a more concrete approach to distributions [66], which he included in applied mathematics courses at the University of Wisconsin. (The innovation was recognized by a Reynolds award in 1956.) It took the mathematical community a while to agree that distributions were useful. This happened only when the theory led to major new develop- ments; see the five books on generalized functions by Gelfand and coauthors [37], and especially the four volumes by H¨ormander [52] on partial differ- ential equations. iii iv PREFACE A detailed description of the now classical material in the present text- book may be found in the introductions to the various chapters. The survey in Chapter 1 mentions work of Euler and Daniel Bernoulli, which preceded the elaborate work of Fourier related to the heat equation. Dirichlet’s rig- orous convergence theory for Fourier series of “good” functions is covered in Chapter 2. The possible divergence in the case of continuous functions is treated, as well as the remarkable Gibbs phenomenon. Chapter 3 shows how such problems were overcome around 1900 by the use of summability methods, notably by Fej´er. Soon thereafter, the notion of square integrable functions in the sense of Lebesgue would lead to an elegant treatment of Fourier series as orthogonal series. However, even summability methods and L2 theory were not general enough to satisfy the needs of applications. Many of these needs were finally met by Schwartz’s distributional theory (Chapter 4). The classical restrictions on many operations, such as differ- entiation and termwise integration or differentiation of infinite series, could be removed. After some general results on metric and normed spaces, including a construction of completion, Chapter 5 discusses inner product spaces and Hilbert spaces. It thus provides the theoretical setting for a good treatment of general orthogonal series and orthogonal bases (Chapter 6). Chapter 7 is devoted to important classical orthogonal systems such as the Legendre polynomials and the Hermite functions. Most of these orthogonal systems arise also as systems of eigenfunctions of Sturm–Liouville eigenvalue prob- lems for differential operators, as shown in Chapter 8. That chapter ends with results on Laplace’s equation (Dirichlet problem) and spherical har- monics. Chapter 9 treats Fourier transformation for well-behaved integrable functions on R. Among the well-behaved functions the Hermite functions stand out; here they appear as eigenfunctions of the linear harmonic oscil- lator in quantum mechanics. At this stage the student should be well-prepared for a general theory of Fourier integrals. The basic questions are to represent larger or unruly functions by trigonometric integrals, and to make Fourier inversion widely possible. A convenient class to work with are the so-called tempered dis- tributions, which include all functions of at most polynomial growth, as well as their (generalized) derivatives of arbitrary order. A good start- ing point to prove unlimited inversion is the observation that the Fourier transform operator commutes with the Hermite operator = x2 D2, where D stands forF differentiation, d/dx. It follows that theH two operators− PREFACE v have the same eigenfunctions. Now the normalized eigenfunctions of are 2 H the Hermite functions hn, which form an orthonormal basis of L . Tem- pered distributions also have a unique representation cnhn; see Chapter 10. The (normalized) Fourier operator transforms the series cnhn into n F P ( i) cnhn, while the reflected Fourier operator R multiplies the expan- − n F P 2 sion coefficients by i . Thus is inverted by R; cf. Chapter 11. For L thisP approach goes back to WienerF [124]. [TheF author has used Hermite se- ries to extend Fourier theory to a much larger class of generalized functions than tempered distributions; see [67], and cf. Zhang [126].] Chapter 12 first deals with one-sided integral transforms such as the Laplace transform, which are important for initial value problems. Next come multiple Fourier transforms. The most important application is to so- called fundamental solutions of certain partial differential equations. In the case of the wave equation one thus obtains the response to a sharply time- limited signal at time zero at the origin. As a striking result one finds that communication governed by that equation works poorly in even dimensions, and works really well only in R3 ! The short final Chapter 13 sketches the theory of general Schwartz dis- tributions and two-sided Laplace transforms. Acknowledgements. Thanks are due to University of Amsterdam colleague Jan van de Craats, who converted my sketches into the nice figures in the text. I also thank former and present colleagues who have encouraged me to write the present “Mathematical Methods” book. Last but not least, I thank the many students who have contributed to the exposition by their questions and comments; it was a pleasure to work with them! Both categories come together in Jan Wiegerinck, who also became a good friend, and director of the Korteweg–de Vries Institute for Mathematics, a nice place to work. Amsterdam, Spring, 2011 Jaap Korevaar Contents Preface iii Chapter 1. Introduction and survey 1 1.1. Power series and trigonometric series 1 1.2. New series by integration or differentiation 4 1.3. Vibrating string and sine series. A controversy 7 1.4. Heat conduction and cosine series 11 1.5. Fourier series 14 1.6. Fourier series as orthogonal series 19 1.7. Fourier integrals 22 Chapter 2. Pointwise convergence of Fourier series 27 2.1. Integrable functions. Riemann–Lebesgue lemma 27 2.2. Partial sum formula. Dirichlet kernel 32 2.3. Theorems on pointwise convergence 35 2.4. Uniform convergence 38 2.5. Divergence of certain Fourier series 41 2.6. The Gibbs phenomenon 44 Chapter 3. Summability of Fourier series 49 3.1. Ces`aro and Abel summability 49 3.2. Ces`aro means. Fej´er kernel 53 3.3. Ces`aro summability: Fej´er’s theorems 56 3.4. Weierstrass theorem on polynomial approximation 58 3.5. Abel summability. Poisson kernel 61 3.6. Laplace equation: circular domains, Dirichlet problem 64 Chapter 4. Periodic distributions and Fourier series 69 4.1. The space L1. Test functions 69 4.2. Periodic distributions: distributions on the unit circle 74 4.3. Distributional convergence 80 4.4. Fourier series 83 vii viii CONTENTS 4.5. Derivatives of distributions 85 4.6. Structure of periodic distributions 91 4.7. Product and convolution of distributions 97 Chapter 5. Metric, normed and inner product spaces 101 5.1. Metrics 101 5.2. Metric spaces: general results 104 5.3. Norms on linear spaces 110 5.4. Normed linear spaces: general results 115 5.5. Inner products on linear spaces 121 5.6. Inner product spaces: general results 125 Chapter 6. Orthogonal expansions and Fourier series 133 6.1. Orthogonal systems and expansions 133 6.2. Best approximation property. Convergence theorem 136 6.3. Parseval formulas. Convergence of expansions 139 6.4. Orthogonalization 144 6.5. Orthogonal bases 149 6.6. Structure of inner product spaces 153 Chapter 7. Classical orthogonal systems and series 157 7.1. Legendre polynomials: Properties related to orthogonality 157 7.2. Other orthogonal systems of polynomials 164 7.3. Hermite polynomials and Hermite functions 170 7.4. Integral representations and generating functions 174 Chapter 8. Eigenvalue problems related to differential equations 181 8.1. Second order equations. Homogeneous case 181 8.2. Non-homogeneous equation. Asymptotics 186 8.3. Sturm–Liouville problems 190 8.4. Laplace equation in R3;polarcoordinates 196 8.5. Spherical harmonics and Laplace series 204 Chapter 9. Fourier transformation of well-behaved functions 213 9.1. Fourier transformation on L1(R) 213 9.2. Fourier inversion 218 9.3. Operations on functions and Fourier transformation 222 9.4. Products and convolutions 225 9.5. Applications in mathematics 228 9.6. The test space and Fourier transformation 233 S CONTENTS ix 9.7. Application: the linear harmonic oscillator 235 9.8. More applications in mathematical physics 237 Chapter 10. Generalized functions of slow growth: tempered distributions 243 10.1. Initial Fourier theory for the class 243 10.2. Fourier transformation in L2(R)P 246 10.3. Hermite series for test functions 250 10.4. Tempered distributions 253 10.5. Derivatives of tempered distributions 257 10.6. Structure of tempered distributions 260 Chapter 11.
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