This is a very comprehensive introduction to the topic, meant for advanced undergraduates, which can be used in a variety of ways. The author explains this in detail: one can teach a one-semester or a two-semester course from the book, or one can use several chapters as auxiliary material for another course. Most instructors will probably teach a one-semester course from the book; in that case, there is an ample choice of topics, especially when it comes to the more applied chapters.
The first three chapters (Fourier representations, Convolutions, and the calculus for finding Fourier transforms on the reals) are part of the core of any course on the subject. After this, the author offers a choice. One can either continue with Fourier transform calculus, namely the discrete Fourier transform, the fast Fourier transform, and operators, or skip these topics and study generalized functions.
After this, there is a wide range of applications, organized in five chapters, which form three blocks. These are partial differential equations, probability, and sampling (this last one discusses wavelets, and musical tones in detail.)
This reviewer (a non-specialist) sees nothing wrong what is covered in this book, and finds the selection of topics interesting. He sometimes has problems with the way in which the material is covered, namely that he is afraid that the target audience, advanced undergraduates, will find the book simply too difficult. Many of them will not understand the first sentence of the preface written to the students, because of the word "sinusoids". The exercises provide a wealth of interesting examples, and so do the chapters on applications. Perhaps it would have been better to place at least some of these examples right next to the theoretical facts that they are meant to illustrate. At one point in the preface, the author encourages the reader to sometimes jump ahead to the applications chapters before continuing with theory; it would have been helpful write the book in a way that such jumping is not necessary.
That said, if the student has what it takes to get through this challenging book, he will clearly have a solid foundation of the subject, and there is a good chance that she will even like it.
Miklós Bóna is Associate Professor of Mathematics at the University of Florida.