This book is a very readable introduction to Fourier series suitable for scientists and engineers. It is sprinkled with hints about more recent developments and has a lot of nice historical comments that will intrigue the best students and math majors. The author almost talks to the readers and skillfully highlights what is important. A fair amount of the material is in the extensive set of exercises. If this very nice text had been available when I was teaching, I would have used it for a junior-senior level course for science and math majors. —Kenneth A. Ross, University of Oregon, Eugene
This is a concise introduction to Fourier series covering history, major themes, theorems, examples and applications. It can be used to learn the subject, and also to supplement, enhance and embellish undergraduate courses on mathematical analysis.
The book begins with a brief summary of the rich history of Fourier series over three centuries. The subject is presented in a way that enables the reader to appreciate how a mathematical theory develops in stages from a practical problem (such as conduction of heat) to an abstract theory dealing with concepts such as sets, functions, infinity and convergence. The abstract theory then provides unforeseen applications in diverse areas.
The author starts out with a description of the problem that led Fourier to introduce his famous series. The mathematical problems this leads to are then discussed rigorously. Examples, exercises and directions for further reading and research are provided, along with a chapter that provides materials at a more advanced level suitable for graduate students. The author demonstrates applications of the theory to a broad range of problems.
The exercises of varying levels of difficulty that are scattered throughout the book will help readers test their understanding of the material.
0. A History of Fourier Series
1. Heat Conduction and Fourier Series
2. Convergence of Fourier Series
3. Odds and Ends
4. Convergence in L2 and L1
5. Some Applications
A. A Note on Normalisation
B. A Brief Bibliography
Bhatia’s textbook, Fourier Series, makes two important claims in its preface — that it will convey a sense of the importance and applicability of Fourier analysis to the reader and that it is accessible, for the most part, to a third year undergraduate student of mathematics. Bhatia attempts to fulfill these assurances by including a section on the sine and cosine functions containing ample examples and graphs, numerous historical references, and a concluding chapter on the applications of Fourier analysis while drawing on the reader’s knowledge of real analysis and differential equations. Bhatia does make references to complex analysis, and chapter four of his text looks more like the second course in real analysis than the first (Bhatia does alert the reader to the more advanced material found in chapter four in his preface), but for the most part the text does seem appropriate for the stated audience.