Early Fourier Analysis

This is a polished introduction to classical Fourier analysis designed for students early in their undergraduate career, perhaps even just after a third term of calculus. The author, well-known number-theorist Hugh Montgomery, says that such students will find in his book “… a gentle introduction to the art of writing proofs and will be better prepared for advanced calculus and complex variables”.

The introductory material occupies a small portion of the book, but it’s notable for the way that Montgomery tries to finesse issues of gaps in a student’s background. He has a quick review of some basic information about series (with proofs), a fast trip through the main theorems of analysis with definitions but no proofs, and an even faster ride through Lebesgue measure theory with theorems, no proofs, and short a few definitions. Montgomery says, “… neither lack of rigor nor absence of advanced training in analysis should interfere with the acquisition of Fourier analysis in its most classical settings”.

The first chapter does complex numbers: first their algebra and geometry, then complex polynomials and power series. As a bonus we get two proofs of the Fundamental Theorem of Algebra.

Fourier analysis starts in earnest with the discrete Fourier transform and roots of unity. It’s a natural place to start since there are no issues about convergence. Besides, from the really applied side, this is the heart and soul. Montgomery gives a short and sweet discussion of the Fast Fourier transform. He explains the key part of the algorithm without getting into any implementation details.

From this point on Montgomery takes the student through the classical Fourier analysis bit by bit, citing results from analysis as needed. First come Fourier series for functions in L¹ and a careful first look convergence questions. Fourier series for functions in L² and Parseval’s identity follow. A chapter on summability looks at both Cesàro and Abel forms, and then proves a couple of tauberian-style theorems. Trigonometric polynomials get a whole chapter to themselves, with a much more extensive treatment than in comparable texts. We see, for example, the Bernstein inequality, Littlewood and Barker polynomials, and some results on the approximation of continuous functions.

There is a full chapter on applications of Fourier series, where “application” means to another area in mathematics, and not necessarily one closely connected to Fourier series. But the heat equation is here, as are the wave equation, some inequalities, Bernoulli polynomials and the uniform distribution. Much of the chapter is more in the vein of “other cool stuff you can do”.

Montgomery finishes with a standard treatment of the Fourier transform for real or complex valued functions, and then a chapter that discusses multidimensional analogs of the discrete FFT, Fourier series, and the Fourier transform.

This is a book that would likely appeal to strong undergraduate mathematics students and give them a real sense of what more advanced analysis is like. But I am skeptical that it would work in a class for most mid-level undergraduates. The book presumes a mathematical maturity and an ability to read mathematics that is uncommon at that level. On the other hand, portions of the book might work very well for a capstone course or independent study.

Montgomery notes that his book has much more content than would work in a single term, and also that he believes it to be appropriate for students at a variety of levels. Indeed, much of the material here would not be out of place for first year graduate students.

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Bill Satzer (wjsatzer@mmm.com) 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.