Fourier Analysis and Hausdorff Dimension
Fourier Analysis and Hausdorff Dimension
$97.99
USDDuring the past two decades there has been active interplay between geometric measure theory and Fourier analysis. This book describes part of that development, concentrating on the relationship between the Fourier transform and Hausdorff dimension. The main topics concern applications of the Fourier transform to geometric problems involving Hausdorff dimension, such as Marstrand type projection theorems and Falconer's distance set problem, and the role of Hausdorff dimension in modern Fourier analysis, especially in Kakeya methods and Fourier restriction phenomena. The discussion includes both classical results and recent developments in the area. The author emphasises partial results of important open problems, for example, Falconer's distance set conjecture, the Kakeya conjecture and the Fourier restriction conjecture. Essentially self-contained, this book is suitable for graduate students and researchers in mathematics.
- Presents cutting-edge results in a very active area of research
- Much of the material appears here for the first time in book form
- Includes open problems that may spur readers on to further research
Reviews & endorsements
'Mattila deserves kudos for having written an excellent text for the community of graduate students and research mathematicians with an analytic bent, one that exposes in considerable detail a particularly rich seam of mathematics at the interface between harmonic analysis and geometric measure theory in Euclidean space … Libraries should be encouraged to buy their copies in haste.' Tushar Das, MAA Reviews
'In addition to a clear, direct writing style, one of the main virtues of this book is the bibliography. (There is a three-page two-column index of authors cited.) Though the book was published in 2015, the author has managed to incorporate references and techniques from many articles that were published as late as 2014. Thus this book is still up to date a few years after its publication. This is an excellent place to begin a study of the interplay between dimension and Fourier transforms.' Benjamin Steinhurst, MathSciNet
Product details
July 2015Adobe eBook Reader
9781316355527
0 pages
0kg
5 b/w illus.
This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
- Preface
- Acknowledgements
- 1. Introduction
- 2. Measure theoretic preliminaries
- 3. Fourier transforms
- 4. Hausdorff dimension of projections and distance sets
- 5. Exceptional projections and Sobolev dimension
- 6. Slices of measures and intersections with planes
- 7. Intersections of general sets and measures
- 8. Cantor measures
- 9. Bernoulli convolutions
- 10. Projections of the four-corner Cantor set
- 11. Besicovitch sets
- 12. Brownian motion
- 13. Riesz products
- 14. Oscillatory integrals (stationary phase) and surface measures
- 15. Spherical averages and distance sets
- 16. Proof of the Wolff–Erdoğan Theorem
- 17. Sobolev spaces, Schrödinger equation and spherical averages
- 18. Generalized projections of Peres and Schlag
- 19. Restriction problems
- 20. Stationary phase and restriction
- 21. Fourier multipliers
- 22. Kakeya problems
- 23. Dimension of Besicovitch sets and Kakeya maximal inequalities
- 24. (n, k) Besicovitch sets
- 25. Bilinear restriction
- References
- List of basic notation
- Author index
- Subject index.
