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Fourier Analysis and Hausdorff Dimension

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: July 2015
  • availability: Available
  • format: Hardback
  • isbn: 9781107107359


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About the Authors
  • 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 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
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    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

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    Product details

    • Date Published: July 2015
    • format: Hardback
    • isbn: 9781107107359
    • length: 452 pages
    • dimensions: 235 x 157 x 29 mm
    • weight: 0.76kg
    • contains: 5 b/w illus.
    • availability: Available
  • Table of Contents

    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
    List of basic notation
    Author index
    Subject index.

  • Author

    Pertti Mattila, University of Helsinki
    Pertti Mattila is Professor of Mathematics at the University of Helsinki and an expert in geometric measure theory. He has authored the book Geometry of Sets and Measures in Euclidean Spaces as well as more than 80 other scientific publications.

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