Lectures on the Fourier Transform and Its Applications

This is a lively introduction to the Fourier integral. It is slanted very much toward signal processing (it grew out of a course EE261 for undergraduate electrical engineering students at Stanford). It is a mathematical treatment, although it omits most of the topics of interest to mathematicians. Thus there is quite a lot about impulse responses, windowing, filters of all types, the Nyquist limit (here referred to as the Nyquist–Shannon sampling theorem), and aliasing, but not much about convergence, function spaces, and the difference between \(L^1\) and \(L^2\) functions. There is a good and early treatment of distributions. The book is mostly about the Fourier transform on the real line but includes detailed coverage of finite Fourier transforms and of \(n\)-dimensional Fourier transforms. It also covers the rudiments of Fourier series, although this is mostly for orientation rather than as a preliminary to the Fourier transform.

The book uses MATLAB for calculations, and there are scripts and data set on box.com, although I was not able to access this using the instructions in the book. The author says on p. 75, “I tend to use Mathematica, and any standard package should be sufficient.”

The problems are wonderful. Each one takes an application area and develops it in some detail. They remind me of the (also excellent) problems in Körner’s Exercises for Fourier Analysis.

In my casual reading, I spotted a lot of typographical errors (9 at last count) without looking very hard; most of these would not be confusing to the student. The book has a good index (with a separate symbol index). There’s no bibliography, although there are many in-line references to other textbooks and research papers.

I like this book a lot. I think it’s a good choice for students who are interested in signal processing, or who could be persuaded to take an interest in signal processing, even though it omits a lot of pure-mathematics topics. A good alternative for pure-math students is Chandrasekharan’s Classical Fourier Transforms.

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.