**University Lecture Series**

Volume: 29;
2003;
137 pp;
Softcover

MSC: Primary 42;
Secondary 28

Print ISBN: 978-0-8218-3449-7

Product Code: ULECT/29

List Price: $37.00

Individual Member Price: $29.60

**Electronic ISBN: 978-1-4704-1837-3
Product Code: ULECT/29.E**

List Price: $37.00

Individual Member Price: $29.60

#### Supplemental Materials

# Lectures on Harmonic Analysis

Share this page *Editors and Authors: *
*Izabella Łaba; Carol Shubin; Thomas H. Wolff*

“There were lots of young analysts who flocked to Chicago in those years, but virtually from the start it was clear that Tom had a special brilliance … Eventually, the mathematical door would open a crack as Tom discovered a new technique, usually of astonishing originality. The end would now be in sight, as [he] unleashed his tremendous technical abilities … Time after time, [Wolff] would pick a central problem in an area and solve it. After a few more results, the field would be changed forever … In the mathematical community, the common and rapid response to these breakthroughs was that they were seen, not just as watershed events, but as lightning strikes that permanently altered the landscape.”

—Peter W. Jones, Yale University

“Tom Wolff was not only a deep thinker in mathematics but
also a technical master.”

—Barry Simon, California Institute of Technology

Thomas H. Wolff was a leading analyst and winner of the Salem and Bôcher
Prizes. He made significant contributions to several areas of harmonic
analysis, in particular to geometrical and measure-theoretic questions related
to the Kakeya needle problem. Wolff attacked the problem with awesome power and
originality, using both geometric and combinatorial ideas. This book provides
an inside look at the techniques used and developed by Wolff. It is based on a
graduate course on Fourier analysis he taught at Caltech.

The selection of the material is somewhat unconventional in that it
leads the reader, in Wolff's unique and straightforward way, through
the basics directly to current research topics. The book demonstrates
how harmonic analysis can provide penetrating insights into deep
aspects of modern analysis. It is an introduction to the subject as a
whole and an overview of those branches of harmonic analysis that are
relevant to the Kakeya conjecture.

The first few chapters cover the usual background material: the Fourier
transform, convolution, the inversion theorem, the uncertainty principle, and
the method of stationary phase. However, the choice of topics is highly
selective, with emphasis on those frequently used in research inspired by the
problems discussed in later chapters. These include questions related to the
restriction conjecture and the Kakeya conjecture, 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, and they continue to stimulate first-rate
work.

The book focuses on laying out a solid foundation for further reading and
research. Technicalities are kept to a minimum, and simpler but more basic
methods are often favored over the most recent methods. The clear style of the
exposition and the quick progression from fundamentals to advanced topics
ensure that both graduate students and research mathematicians will benefit
from the book.