Lectures on Harmonic Analysis
About this Title
Thomas H. Wolff. Edited by Izabella Łaba, University of British Columbia, Vancouver, BC, Canada and Carol Shubin, California State University Northridge, Northridge, CA
Publication: University Lecture Series
Publication Year: 2003; Volume 29
ISBNs: 978-0-8218-3449-7 (print); 978-1-4704-1837-3 (online)
MathSciNet review: MR2003254
MSC: Primary 42-02; Secondary 42B15
This book demonstrates how harmonic analysis can provide penetrating insights into deep aspects of modern analysis. It is both 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 usual background material is covered in the first few chapters: 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 the 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 ensures that both graduate students and research mathematicians will benefit from the book.
Graduate students and research mathematicians interested in harmonic analysis.
Table of Contents
- Chapter 1. The $L^1$ Fourier transform
- Chapter 2. The Schwartz space
- Chapter 3. Fourier inversion and the Plancherel theorem
- Chapter 4. Some specifics, and $L^p$ for $p<2$
- Chapter 5. The uncertainty principle
- Chapter 6. The stationary phase method
- Chapter 7. The restriction problem
- Chapter 8. Hausdorff measures
- Chapter 9. Sets with maximal Fourier dimension and distance sets
- Chapter 10. The Kakeya problem
- Chapter 11. Recent work connected with the Kakeya problem
- Historical notes