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  3. Fourier Analysis and Hausdorff Dimension
  • Cited by 95
Publisher:
Cambridge University Press
Online publication date:
September 2015
Print publication year:
2015
Online ISBN:
9781316227619
DOI:
https://doi.org/10.1017/CBO9781316227619
Subjects:
Mathematics (general), Mathematics, Abstract Analysis, Real and Complex Analysis
Series:
Cambridge Studies in Advanced Mathematics (150)
Information

Book description

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.

Reviews

'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 Source: 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 Source: MathSciNet

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Contents

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Contents

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References
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