This is a good Fourier analysis textbook for upper level undergraduates and for lower level graduate students. The numerous applications in the book make it appealing to engineers and both applied and pure mathematicians who feel the need to justify their existence.
The topics included in the book are quite extensive, including boundary value problems, spaces, Sturm-Liouville problems, convolution and the delta function, Fourier transforms and integrals, and special topics such as DFT, FFT, and filtering, and wavelets. This book certainly has plenty of material for a two semester sequence in Fourier analysis.
Stade incorporates tidbits about the history of Fourier analysis throughout the text by not only including when people studied the material but also why they were interested in it. This gives the subject some context and perspective that is quite refreshing to see in a mathematics textbook — so often the historical material is only incorporated in lower level textbooks.
Throughout the text, Stade uses DIY (Do It Yourself) to indicate places where he has skipped steps to get a result. Although this phrase will not necessarily be popular among students, it is less pompous than statements like “it can easily be shown,” which students tend to find frustrating when they can’t “easily see it.” In addition, he has included plenty of examples throughout the text and often refers to them in the exercises. The exercises are very straightforward and sometimes even include hints, which will certainly be popular with students.
The author does a great job of incorporating his sense of humor throughout this book. (He has given a talk entitled “Joseph Fourier and the Birth of Disco” several times. Although no mention of disco is made in this text, Jimi Hendrix does make an appearance.) This makes the book more enjoyable. As faculty we often complain that the students don’t read their books, but then we often supply them with dry material. Stade went to great lengths to include humor—even poking fun at himself:
Evaluation of infinite series is a sport enjoyed by many people. Number theorists, of whom the author is one (as he will admit freely in most social situations), are especially keen on this sport, because of the relevance of these series to the behavior of prime numbers and to other, more or less (or much less) related, phenomena. Others take part because infinite series and their values have applications in chemistry, in physics, and so on. But like all sports, the primary and best reason for participating is just for fun. (If anyone knows how to have fun, it’s the number theorists, we can assure you.)
Sharon Schaffer Vestal is Associate Professor of Mathematics at Missouri Western State University in Saint Joseph, Missouri.