This book is based on a graduate course in Fourier analysis taught by
Tom Wolff in the Spring of 2000 at the California Institute of Technology.
Tom wrote up a set of notes which he distributed to his students and made
widely available on the Internet. His intention was to publish these notes in
a book format. After Tom’s untimely death on July 31, 2000, I was asked
to complete this work.
The selection of the material is somewhat unconventional in that the
book leads us, in Tom’s unique and straightforward way, through the basics
directly to current research topics. Chapters 1-4 cover standard background
material: the Fourier transform, convolution, the inversion theorem, the
Hausdorff-Young inequality. Chapters 5 and 6 introduce the uncertainty
principle and the stationary phase method. The choice of topics is highly
selective, with emphasis on those frequently useful in research inspired by
the problems discussed in the remaining chapters. The latter include ques-
tions related to the restriction and Kakeya conjectures, distance sets, and
Fourier transforms of singular measures. These problems are diverse but
often interconnected; they all combine sophisticated Fourier analysis with
intriguing links to other areas of mathematics (combinatorics, number the-
ory, partial differential equations); and they continue to stimulate first-rate
work. This book focuses on laying out a solid foundation for further read-
ing and, hopefully, research. Technicalities are kept down to the necessary
minimum, and simpler but more basic methods are often favoured over the
most recent ones.
The book is intended for all mathematical audiences a novice and
an expert may read it on different levels, but both should be able to find
something of interest to them. A background in harmonic analysis is not
necessary. Some mathematical maturity, however, will be helpful; the more
junior readers should expect to work hard and to be rewarded generously
for their efforts.
Tom’s original manuscript constitutes Chapters 1-9 of this book. I have
edited this part, clarifying a number of points and correcting typos and
small errors. Most of the changes are quite minor, with the exception of
Chapter 9A which was considerably expanded at the request of many read-
ers of the original version. Chapter 10 is based on Burak Erdogan’s notes of
Tom’s Caltech lectures; I am responsible for its final shape. The last part of
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