This is a companion text for the author’s

*Fourier Analysis*. Both are very erudite books and cover a wide variety of topics usually not seen in Fourier analysis books, ranging over many parts of mathematics and physics. This is a good book for browsing, as the prerequisite for most problems are not very high (and sometimes do not even include previous knowledge of Fourier analysis). No solutions are given, but most problems are to prove something and enough details are given that you can at least spot the outline of the proof.

The present book might be best described as “further investigations suggested by the first book,” and most chapters of the first book has a corresponding chapter in this book with related exercises. For example, the first chapter of Fourier Analysis uses the idea of Cesàro summability for Fourier series, and the first chapter of the present book deals with Toeplitz matrices and further kinds of summability and their applications. The author emphasizes in the Preface (p. ix) that he has tried “to produce exercises and not problems. You should find them more in the nature of a hill top walk than a rock climbing expedition.” None of the exercises is drill, and many require considerable effort (although usually not great ingenuity, because there are many hints). There are still many intricate exercises, for example a proof (Chapter 43) of Apéry’s result that the Riemann zeta function value ζ(3) is irrational, but these are broken down into many small steps.

The exercises are usually not in the direct line of development and are often tangential, sometimes extremely tangential. For example, Chapter 54 in the main book is a brief biography of Lord Kelvin and his role as a successor of Fourier. The corresponding chapter of the exercise book consists of two proofs of the infinitude of the primes, one by Euclid and one by Pólya. What’s the connection? I think it is this sentence from the main book: “A novel always retains the signature of its author, a proof in pure mathematics may still delight us after a hundred (consider Liouville’s proof given as Theorem 38.2 of the existence of transcendental numbers) or a thousand years (consider Euclid’s proof that there exist an infinity of primes), but the physics breakthrough of today is the common sense of tomorrow.” (p. 271)

As you would expect, many of the exercises deal with the partial differential equations of physics, but many other topics are also covered, including such unlikely ones as a great deal of number theory (including Dirichlet’s theorem on primes in arithmetic progression and the Prime Number Theorem), some complex analysis (including the Hadamard gap theorem), and some probability and statistics (including chi-square tests and the Kolmogorov 0–1 theorem). This book, like the main one, dips its toe into some non-classical topics such as the discrete Fourier transform, wavelets, and Fourier analysis on groups, and applied areas such as signal processing.