Fourier analysis and Hausdorff dimension / Pertti Mattila.
By: Mattila, Pertti [author].
Series: Cambridge studies in advanced mathematics ; 150.Publisher: Cambridge : Cambridge University Press, 2015Description: xiv, 440 p. : illustrations ; 24 cm.ISBN: 9781107107359.Subject(s): Fourier transformations  Measure theory  Mathematical analysisDDC classification: 515.2433Item type  Current location  Call number  Status  Date due  Barcode  Item holds  

Books 
ISI Library, Kolkata

515.2433 W444 (Browse shelf)  Checked out  03/04/2020  136865 
Browsing ISI Library, Kolkata Shelves Close shelf browser
Includes bibliographical references and indexes.
1. Introduction 
Part 1. Preliminaries and some simpler applications of the Fourier transform.
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 
Part 2. Specific constructions.
8. Cantor measures 
9. Bernoulli convolutions 
10. Projections of the fourcorner Cantor set 
11. Besicovitch sets 
12. Brownian motion 
13. Riesz products 
14. Oscillatory integrals (stationary phase) and surface measures 
Part 3. Deeper applications of the Fourier transform.
15. Spherical averages and distance sets 
16. Proof of the WolffErdogan Theorem 
17. Sobolev spaces, Schrödinger equation and spherical averages 
18. Generalized projections of Peres and Schlag 
Part 4. Fourier restriction and Kakeya type problems. 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 
Indexes.
"During 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 selfcontained, this book is suitable for graduate students and researchers in mathematics."Back cover.
There are no comments for this item.