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Fourier analysis and Hausdorff dimension

Mattila, Pertti.

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Fourier analysis and Hausdorff dimension

Record Type:	Electronic resources : Monograph/item
Title/Author:	Fourier analysis and Hausdorff dimension/ Pertti Mattila.
remainder title:	Fourier Analysis & Hausdorff Dimension
Author:	Mattila, Pertti.
Published:	Cambridge :Cambridge University Press, : 2015.,
Description:	xiv, 440 p. :ill., digital ;24 cm.
Notes:	Title from publisher's bibliographic system (viewed on 05 Oct 2015).
[NT 15003449]:	Preface -- Acknowledgements -- Introduction -- Part 1. Preliminaries and some simpler applications of the Fourier transform. Measure theoretic preliminaries -- Fourier transforms -- Hausdorff dimension of projections and distance sets -- Exceptional projections and Sobolev dimension -- Slices of measures and intersections with planes -- Intersections of general sets and measures -- Part 2. Specific constructions. Cantor measures -- Bernoulli convolutions -- Projections of the four-corner Cantor set -- Besicovitch sets -- Brownian motion -- Riesz products -- Oscillatory integrals (stationary phase) and surface measures -- Part 3. Deeper applications of the Fourier transform. Spherical averages and distance sets -- Proof of the Wolff-Erdogan Theorem -- Sobolev spaces, Schrodinger equation and spherical averages -- Generalized projections of Peres and Schlag -- Part 4. Fourier restriction and Kakeya type problems. Restriction problems -- Stationary phase and restriction -- Fourier multipliers -- Kakeya problems -- Dimension of Besicovitch sets and Kakeya maximal inequalities -- (n, k) Besicovitch sets -- Bilinear restriction.
Subject:	Fourier transformations. -
Online resource:	https://doi.org/10.1017/CBO9781316227619
ISBN:	9781316227619

Fourier analysis and Hausdorff dimension
Mattila, Pertti.

Fourier analysis and Hausdorff dimension[electronic resource] /Fourier Analysis & Hausdorff DimensionPertti Mattila. - Cambridge :Cambridge University Press,2015. - xiv, 440 p. :ill., digital ;24 cm. - Cambridge studies in advanced mathematics ;150. - Cambridge studies in advanced mathematics ;150..

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Preface -- Acknowledgements -- Introduction -- Part 1. Preliminaries and some simpler applications of the Fourier transform. Measure theoretic preliminaries -- Fourier transforms -- Hausdorff dimension of projections and distance sets -- Exceptional projections and Sobolev dimension -- Slices of measures and intersections with planes -- Intersections of general sets and measures -- Part 2. Specific constructions. Cantor measures -- Bernoulli convolutions -- Projections of the four-corner Cantor set -- Besicovitch sets -- Brownian motion -- Riesz products -- Oscillatory integrals (stationary phase) and surface measures -- Part 3. Deeper applications of the Fourier transform. Spherical averages and distance sets -- Proof of the Wolff-Erdogan Theorem -- Sobolev spaces, Schrodinger equation and spherical averages -- Generalized projections of Peres and Schlag -- Part 4. Fourier restriction and Kakeya type problems. Restriction problems -- Stationary phase and restriction -- Fourier multipliers -- Kakeya problems -- Dimension of Besicovitch sets and Kakeya maximal inequalities -- (n, k) Besicovitch sets -- Bilinear restriction.

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.

ISBN: 9781316227619Subjects--Topical Terms:

533900
Fourier transformations.

LC Class. No.: QA403.5 / .M38 2015

Dewey Class. No.: 515.2433

Fourier analysis and Hausdorff dimension
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100 1 $a Mattila, Pertti. $3 522209
245 1 0 $a Fourier analysis and Hausdorff dimension $h [electronic resource] / $c Pertti Mattila.
246 3 $a Fourier Analysis & Hausdorff Dimension
260 $a Cambridge : $b Cambridge University Press, $c 2015.
300 $a xiv, 440 p. : $b ill., digital ; $c 24 cm.
490 1 $a Cambridge studies in advanced mathematics ; $v 150
500 $a Title from publisher's bibliographic system (viewed on 05 Oct 2015).
505 0 $a Preface -- Acknowledgements -- Introduction -- Part 1. Preliminaries and some simpler applications of the Fourier transform. Measure theoretic preliminaries -- Fourier transforms -- Hausdorff dimension of projections and distance sets -- Exceptional projections and Sobolev dimension -- Slices of measures and intersections with planes -- Intersections of general sets and measures -- Part 2. Specific constructions. Cantor measures -- Bernoulli convolutions -- Projections of the four-corner Cantor set -- Besicovitch sets -- Brownian motion -- Riesz products -- Oscillatory integrals (stationary phase) and surface measures -- Part 3. Deeper applications of the Fourier transform. Spherical averages and distance sets -- Proof of the Wolff-Erdogan Theorem -- Sobolev spaces, Schrodinger equation and spherical averages -- Generalized projections of Peres and Schlag -- Part 4. Fourier restriction and Kakeya type problems. Restriction problems -- Stationary phase and restriction -- Fourier multipliers -- Kakeya problems -- Dimension of Besicovitch sets and Kakeya maximal inequalities -- (n, k) Besicovitch sets -- Bilinear restriction.
520 $a 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.
650 0 $a Fourier transformations. $3 533900
650 0 $a Measure theory. $3 516951
650 0 $a Mathematical analysis. $3 516833
830 0 $a Cambridge studies in advanced mathematics ; $v 150. $3 3470470
856 4 0 $u https://doi.org/10.1017/CBO9781316227619