This is a concise and very focused introduction to the theory of Fourier transforms of functions defined on the real line. The prerequisites are a modest grasp of Lebesgue integration (mostly the convergence theorems); it’s also helpful to be familiar with Fourier series, as the book uses these for motivation in some cases.

This is primarily a pure-math view of the subject. It does show how Fourier transforms can be used to show the existence and uniqueness of solutions to two heat equations (differential equations), but does not show how to solve them. It also gives applications to other pure-math areas, such as the central limit theorem and Tauberian theorems.

The first half of the book (Chapter 1) deals with the theory for \(L^1\) functions. The theory is most straightforward in this case, and many of the uses of Fourier transforms are for \(L^1\) functions. The main drawback is that the transform of an \(L^1\) function may not be \(L^1\), so the transform is not always invertible. Most of the rest of the book (Chapter 2) deals with \(L^2\) functions; the theory is more symmetric here because the transforms are also \(L^2\) functions, but it is harder to get started because the naive approach of defining the transform as an integral doesn’t always converge for \(L^2\) functions, and the definition has to be bootstrapped. This chapter depends heavily on the \(L^1\) theory. The rest of the book (Chapter 3) looks briefly at Fourier-Stieltjes transforms, a generalization of the \(L^1\) theory.

I like this book because it is modern without being abstract. The “classical” in the title means both that it sticks to to real line, and that it (generally) avoids the Banach algebra approach. Most treatments of Fourier transforms are aimed at the applications in physics, and do not go into as much depth into the theory as this book does. Another good but extremely concise treatment (that does include Banach algebras) is in Rudin’s *Real and Complex Analysis*. An older book that goes into even more depth is Titchmarsh’s *Introduction to The Theory of Fourier Integrals*. Wiener’s *The Fourier Integral and Certain of its Applications* has a lot of overlap with this book, but is not as balanced and focuses on Wiener’s research interests.

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.