Fourier Analysis and Its Applications (Graduate Texts in Mathematics, Vol. 223) (Graduate Texts in Mathematics (223))

I have waited for some professional mathematician to review this book, since I think it is such a good book. This has not happened, so here is my un-professional review: I am a retired electrical engineer, and the last three years I have worked on refreshing my old and forgotten school math, and also to going a bit further than during school. When I bought this book I had the following wishes, all of whom it fulfilled brilliantly: 1. It should be usable for self study. 2. It should refresh my fourier series and transforms. 3. It should put these things in context, by presenting physics problems and also by widening the scope to wet the appetite for functional analysis and partial diff eqs. 4. It should do something for my sorely lacking "mathematical maturity". All of these things it did and I just want to add that Mr. Vretblad is an outstanding pedagogue and that the book contains very few typos. (I found two or three in the whole book, none of them important). Contents......Page 10 Preface......Page 8 1.1 The classical partial differential equations......Page 14 1.2 Well-posed problems......Page 16 1.3 The one-dimensional wave equation......Page 18 1.4 Fourier’s method......Page 22 2.1 Complex exponentials......Page 28 2.2 Complex-valued functions of a real variable......Page 30 2.3 Cesàro summation of series......Page 33 2.4 Positive summation kernels......Page 35 2.5 The Riemann–Lebesgue lemma......Page 38 2.6 *Some simple distributions......Page 40 2.7 *Computing with δ......Page 45 3.1 The Laplace transform......Page 52 3.2 Operations......Page 55 3.3 Applications to differential equations......Page 60 3.4 Convolution......Page 66 3.5 *Laplace transforms of distributions......Page 70 3.6 The Z transform......Page 73 3.7 Applications in control theory......Page 80 Summary of Chapter 3......Page 83 4.1 Definitions......Page 86 4.2 Dirichlet’s and Fejér’s kernels; uniqueness......Page 93 4.3 Differentiable functions......Page 97 4.4 Pointwise convergence......Page 99 4.5 Formulae for other periods......Page 103 4.6 Some worked examples......Page 104 4.7 The Gibbs phenomenon......Page 106 4.8 *Fourier series for distributions......Page 109 Summary of Chapter 4......Page 113 5.1 Linear spaces over the complex numbers......Page 118 5.2 Orthogonal projections......Page 123 5.3 Some examples......Page 127 5.4 The Fourier system is complete......Page 132 5.5 Legendre polynomials......Page 136 5.6 Other classical orthogonal polynomials......Page 140 Summary of Chapter 5......Page 143 6.1 The solution of Fourier’s problem......Page 150 6.2 Variations on Fourier’s theme......Page 152 6.3 The Dirichlet problem in the unit disk......Page 161 6.4 Sturm–Liouville problems......Page 166 6.5 Some singular Sturm–Liouville problems......Page 172 Summary of Chapter 6......Page 173 7.1 Introduction......Page 178 7.2 Definition of the Fourier transform......Page 179 7.3 Properties......Page 181 7.4 The inversion theorem......Page 184 7.5 The convolution theorem......Page 189 7.6 Plancherel’s formula......Page 193 7.7 Application 1......Page 195 7.8 Application 2......Page 198 7.9 Application 3: The sampling theorem......Page 200 7.10 *Connection with the Laplace transform......Page 201 7.11 *Distributions and Fourier transforms......Page 203 Summary of Chapter 7......Page 205 8.1 History......Page 210 8.2 Fuzzy points – test functions......Page 213 8.3 Distributions......Page 216 8.4 Properties......Page 219 8.5 Fourier transformation......Page 226 8.6 Convolution......Page 231 8.7 Periodic distributions and Fourier series......Page 233 8.8 Fundamental solutions......Page 234 8.9 Back to the starting point......Page 236 Summary of Chapter 8......Page 237 9.1 Rearranging series......Page 240 9.2 Double series......Page 243 9.3 Multi-dimensional Fourier series......Page 246 9.4 Multi-dimensional Fourier transforms......Page 249 A: The ubiquitous convolution......Page 252 B: The discrete Fourier transform......Page 256 C.1 Laplace transforms......Page 260 C.2 Z transforms......Page 263 C.3 Fourier series......Page 264 C.4 Fourier transforms......Page 265 C.5 Orthogonal polynomials......Page 267 D: Answers to selected exercises......Page 270 E: Literature......Page 278 E......Page 280 P......Page 281 Z......Page 282 TheclassicaltheoryofFourierseriesandintegrals,aswellasLaplacetra- forms, is of great importance for physical and technical applications, and its mathematical beauty makes it an interesting study for pure mathema- cians as well. I have taught courses on these subjects for decades to civil engineeringstudents,andalsomathematicsmajors,andthepresentvolume can be regarded as my collected experiences from this work. There is, of course, an unsurpassable book on Fourier analysis, the tr- tise by Katznelson from 1970. That book is, however, aimed at mathem- ically very mature students and can hardly be used in engineering courses. Ontheotherendofthescale,thereareanumberofmore-or-lesscookbo- 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?eld. There do exist a few texts that ful?ll 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. "This book presents the basic ideas in Fourier analysis and its applications to the study of partial differential equations. It also covers the Laplace and Zeta transformations and the fundaments of their applications. The author has intended to make his exposition accessible to readers with a limited background, for example, those not acquainted with the Lebesgue integral or with analytic functions of a complex variable. At the same time, he has included discussions of more advanced topics such as the Gibbs phenomenon, distributions, Sturm-Liouville theory, Cesaro summability and multi-dimensional Fourier analysis, topics which one usually will not find in books at this level. Many of the chapters end with a summary of their contents, as well as a short historical note. The text contains a great number of examples, as well as more than 350 exercises. In addition, one of the appendices is a collection of the formulas needed to solve problems in the field. Anders Vretblad is Senior Lecturer of Mathematics at Uppsala University, Sweden."--Publisher's website

This book presents the basic ideas in Fourier analysis and its applications to the study of partial differential equations. It also covers the Laplace and Zeta transformations and the fundaments of their applications. The author has intended to make his exposition accessible to readers with a limited background, for example, those not acquainted with the Lebesgue integral or with analytic functions of a complex variable. At the same time, he has included discussions of more advanced topics such as the Gibbs phenomenon, distributions, Sturm-Liouville theory, Cesaro summability and multi-dimensional Fourier analysis, topics which one usually will not find in books at this level.

Many of the chapters end with a summary of their contents, as well as a short historical note. The text contains a great number of examples, as well as more than 350 exercises. In addition, one of the appendices is a collection of the formulas needed to solve problems in the field.

Anders Vretblad is Senior Lecturer of Mathematics at Uppsala University, Sweden.

This book presents the basic ideas in Fourier analysis and its applications to the study of partial differential equations. It also covers the Laplace and Zeta transformations and the fundaments of their applications. The author has intended to make his exposition accessible to readers with a limited background, for example, those not acquainted with the Lebesque integral or with analytic functions of a complex variable. At the same time, he has included discussions of more advanced topics such as the Gibbs phenomenon, distributions, Sturm-Liouville theory, Cesaro summability, and multidimensional Fourier analysis, topics that one usually will not find in books at this level. Many of the chapters end with a summary of their contents, as well as a short historical note. The text contains a a great number of examples, as well as more than 350 exercises. In addition, one of the appendices is a collection of the formulas needed to solve problems in the field. Anders Vretblad is Senior Lecturer of Mathematics at Uppsala University Sweden. A carefully prepared account of the basic ideas in Fourier analysis and its applications to the study of partial differential equations. The author succeeds to make his exposition accessible to readers with a limited background, for example, those not acquainted with the Lebesgue integral. Readers should be familiar with calculus, linear algebra, and complex numbers. At the same time, the author has managed to include discussions of more advanced topics such as the Gibbs phenomenon, distributions, Sturm-Liouville theory, Cesaro summability and multi-dimensional Fourier analysis, topics which one usually does not find in books at this level. A variety of worked examples and exercises will help the readers to apply their newly acquired knowledge. In this introductory chapter, we give a brief survey of three main types of partial differential equations that occur in classical physics.