Fourier Analysis and Its Applications

1. Front Matter

   Pages i-xi

   PDF

2. Introduction

   • Anders Vretblad

   Pages 1-13

3. Preparations

   • Anders Vretblad

   Pages 15-37

4. Laplace and Z transforms

   • Anders Vretblad

   Pages 39-72

5. Fourier series

   • Anders Vretblad

   Pages 73-103

6. L ² Theory

   • Anders Vretblad

   Pages 105-135

7. Separation of variables

   • Anders Vretblad

   Pages 137-163

8. Fourier transforms

   • Anders Vretblad

   Pages 165-195

9. Distributions

   • Anders Vretblad

   Pages 197-226

10. Multi-dimensional Fourier analysis

    • Anders Vretblad

    Pages 227-237

11. Back Matter

    Pages 239-269

    PDF

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.

• Fourier series
• Fourier transform
• calculus
• differential equation
• discrete Fourier transform

From the reviews:

"This book is one in the Graduate Texts in Mathematics series published by Springer. … There is a variety of worked examples as well as 350-plus exercises … . The book is a valuable addition to the literature on Fourier analysis. It is written with more mathematical rigour than many texts … without being totally opaque to the non-specialist. … The examples at the end of each chapter are well structured and a reader working through most of them will achieve a good understanding of the topics." (Graham Brindley, The Mathematical Gazette, Vol. 90 (517), 2006)

"The author … presents the results of his experiences and choices for decades of teaching courses. … The tables and formulas collected … are of great service. At the end of each chapter there is a summary section that discusses the results, gives some history, and suggests instructive exercises. We thus have a solid course on Fourier analysis and its applications interesting for students and specialists in engineering as well as for mathematicians. … I believe that the book will find numerous interested readers." (Elijah Liflyand, Zentralblatt MATH, Vol. 1032 (7), 2004)

"This book is an interesting mixture of a traditional approach … and a more modern one, emphasizing the role of (tempered) distributions and the application aspects of Fourier analysis. … The book is certainly highly recommendable for those who want to learn the essence of Fourier analysis in a mathematically correct way without having to go through too much technical details." (H.G. Feichtinger, Monatshefte für Mathematik, Vol. 143 (2), 2004)

"The book is appropriate for an advanced undergraduate or a master’s level one-term introductory course on Fourier series with applications to boundary value problems. … a deep idea is presented in a non-rigorous way both to show the usefulness of the idea and to stimulate interest in further study. … The book has a goodcollection of exercises … . Each chapter ends with both a summary of its main results and methods and historical notes." (Colin C. Graham, Mathematical Reviews, Issue 2004 e)

• Mathematics Department, San Francisco State University, San Francisco, USA

  S. Axler

• Mathematics Department East Hall, University of Michigan, Ann Arbor, USA

  F. W. Gehring

• Mathematics Department, University of California, Berkeley, USA

  K. A. Ribet

• Department of Mathematics, Uppsala University, Uppsala, Sweden

  Anders Vretblad

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

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