The Fourier Integral and Certain of its Applications

This book is a monograph on several specific topics where Wiener had made major new contributions to harmonic analysis. This book was published in 1933 by Cambridge, reprinted in 1958 by Dover, then reprinted again by Cambridge with a new historical Foreword by J.-P. Kahane. The content was given as a series of lectures at Cambridge in 1932. Wiener’s exposition carried many new ideas and changed the way the subject was viewed, so the book is very important historically.

The centerpiece of the book is Wiener’s General Tauberian Theorem, with Chapter II stating and proving the theorem and Chapter III giving example Tauberian theorems that can be proved using it. These are preceded by an Introduction that develops all the facts that will be needed about the Lebesgue integral, and Chapter I that develops Plancherel’s theory of Fourier transforms in \(L^2\). The General Tauberian Theorem belongs to \(L^1\), and the Fourier transform for \(L^1\) functions is developed in the beginning of Chapter II. Chapter IV is devoted to a different subject, generalized harmonic analysis, including almost-periodic functions.

The exposition of the General Tauberian Theorem is still fresh today, and in fact Choimet & Queffélec’s 2015 book Twelve Landmarks of Twentieth-Century Analysis presents a streamlined version of the same proof. In another sense it has been subsumed into Gelfand’s theory of Banach algebras, of which some of the ideas are present in Wiener’s version (Choimet & Queffélec also cover this approach, for comparison).

This is probably not the best book to learn about any of these topics today. When the book was written, the Lebesgue integral was still not in the mainstream and a lot of the book is devoted to explaining it. Wiener was a pioneer in the areas discussed in the book and they are much more developed today, with good expositions in more recent books. The above-mentioned book by Choimet & Queffélec has an excellent exposition of the Tauberian part. Wiener’s book was not intended as a treatise on Fourier transforms, but to show important uses of them. There are many good books today with a modern view of Fourier transforms; I like Chandrasekharan’s Classical Fourier Transforms for its concreteness and coverage, and Rudin’s Real and Complex Analysis for a slick modern treatment. There are whole books on Tauberian theorems, notably Korevaar’s Tauberian Theory.

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.