This is a wonderful book. The author modestly calls it a “shop window for some of the ideas, techniques and elegant results of Fourier analysis”. It consists of a series of chapters — short interlinked essays, really — accessible to advanced undergraduates. As a text, it excels in the breadth of its treatment but freely sacrifices generality for clarity and readability. The level of rigor varies considerably. (The author approvingly quotes Aristotle: “It is the mark of the educated mind to use for each subject the degree of exactness which it admits.”) There are scads of anecdotes and pieces of history, yet the author urges students to consult original sources because, as a rule, “mathematicians are remarkably incompetent historians.” More than anything, this is just fun to read, to browse, to study.
The work begins with Fejér’s theorem and two additional results: a continuous function on the unit circle is uniquely determined by its Fourier coefficients, and the trigonometric polynomials are uniformly dense in the continuous functions on the unit circle. These results and the method of proof of Fejér’s theorem are the core of the book. In a sense, everything else builds on that.
There are 110 short chapters — each about five pages long. The main sections of the book are titled: Fourier Series, Some Differential Equations, Orthogonal Series, Fourier Transforms, Further Developments and Other Directions. The first four are adequately self-explanatory and the latter two are full of applications and consequences of the earlier material. There are compelling digressions throughout.
The combination of hard analysis with digressions and tangents is consistently delightful. The section on Fourier series discusses the polynomial approximation theorems, behavior at points of discontinuity and convergence theorems, but it also takes up compass and tides, Monte Carlo methods, and Brownian motion. In the midst of the big section on Fourier transforms, the reader meets the heat equation and then immediately is introduced to Lord Kelvin and an extended discussion of the controversy over the age of the earth. Later in this section the author introduces the wave equation and then digresses (in a natural way) to discuss the transatlantic cable and the invention of the mirror galvanometer. In this application “half a million pounds was being staked on the correctness of the solution of a partial differential equation.”
The final two sections (about two hundred pages) are chock full of fascinating digressions, all within the theme of Fourier analysis. These include an extended discussion of “an example of an outstanding statistical treatment” — an analysis of Cyril Burt’s data on race and IQ. Also to be found : a short historical bit on Fourier, lots of questions related to computations (why, what and how do we compute Fourier transforms and how fast can we multiply), a bit of group theory and Fourier analysis on finite Abelian groups, and a discussion of primes in arithmetical progression, just for example.
There are no exercises. For those there is a companion volume, Exercises in Fourier Analysis, which is no less delightful. Fourier Analysis is literate, lively and a true classic. I highly recommend it.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.