This book is an adaptation of the author’s 1996 work, *Applicazioni dell’Analisi Armonica*. This version is roughly twice as long as the original and – more importantly for most of the target audience – is written in English rather than Italian. The goal of this text is to introduce readers to the topic of harmonic analysis, its history, and its applications. Throughout the book, these three major concerns (history, mathematics, and application) are artfully intertwined to form a single, unified narrative. This is not simply a book about mathematics, or even the history of mathematics; it is a story about how the discipline has been applied (to borrow Fourier’s expression) to “the public good and the explanation of natural phenomena.”

As one would expect, the book begins with a fairly substantial biography of Jean-Baptiste Fourier, the father of harmonic analysis. It continues with a discussion of the two main problems that motivated the development of the subject, the vibration of a string and the diffusion of heat. During the course of the book, the author considers a remarkably broad collection of applications, from space exploration to classical music to CAT scans. The treatment of each topic is thorough and comprehensive, paying equal attention to mathematics, application, and historical context.

*The Evolution of Applied Harmonic Analysis* is appropriate for quite a general audience. Anyone familiar with calculus will probably be able to understand most of the subject matter. The book is most suitable for readers who are mathematically and scientifically literate, yet who are not already familiar with the topics being discussed. Nonetheless, the book should also be interesting to professional mathematicians – even to serious students of harmonic analysis – although such readers will doubtless want to skip over some of the more elementary explanations.

It is evident that a great deal of effort has been spent in the preparation of this book. The author has diligently worked to create a cogent, sensibly organized exposition. The text is supplemented with dozens of interesting pictures and graphs. The author certainly deserves to be commended for writing in English, which (as she mentions in the preface) is not her native tongue. Unfortunately, there are a few places in the text where the sentence structure is slightly awkward, and even a bit confusing. In spite of such minor defects, this book constitutes a significant addition to the library of popular mathematical works, and a valuable resource for students of mathematics.

Christopher Hammond is Assistant Professor of Mathematics at Connecticut College.