Fourier Analysis and Convexity

by BRANDOLINI, LUCA

• ISBN13: 9780817632632
• ISBN10: 0817632638
• Format: Hardcover
• Copyright: 2004-11-01
• Publisher: Birkhauser

Summary

Over the course of the last century, the systematic exploration of the relationship between Fourier analysis and other branches of mathematics has lead to important advances in geometry, number theory, and analysis, stimulated in part by Hurwitz's proof of the isoperimetric inequality using Fourier series. This unified, self-contained volume is dedicated to Fourier analysis, convex geometry, and related topics. Specific topics covered include:* the geometric properties of convex bodies* the study of Radon transforms* the geometry of numbers* the study of translational tilings using Fourier analysis* irregularities in distributions* Lattice point problems examined in the context of number theory, probability theory, and Fourier analysis* restriction problems for the Fourier transform The book presents both a broad overview of Fourier analysis and convexity as well as an intricate look at applications in some specific settings; it will be useful to graduate students and researchers in harmonic analysis, convex geometry, functional analysis, number theory, computer science, and combinatorial analysis. A wide audience will benefit from the careful demonstration of how Fourier analysis is used to distill the essence of many mathematical problems in a natural and elegant way.Contributors: J. Beck, C. Berenstein, W.W.L. Chen, B. Green, H. Groemer, A. Koldobsky, M. Kolountzakis, A. Magyar, A.N. Podkorytov, B. Rubin, D. Ryabogin, T. Tao, G. Travaglini, A. Zvavitch

Table of Contents

Preface	v
Lattice Point Problems: Crossroads of Number Theory, Probability Theory and Fourier Analysis	1	(36)
Jozsef Beck
Totally Geodesic Radon Transform of Lp-Functions on Real Hyperbolic Space	37	(22)
Carlos A. Berenstein
Boris Rubin
Fourier Techniques in the Theory of Irregularities of Point Distribution	59	(24)
W.W.L. Chen
Spectral Structure of Sets of Integers	83	(14)
Ben Green
100 Years of Fourier Series and Spherical Harmonics in Convexity	97	(22)
H. Groemer
Fourier Analytic Methods in the Study of Projections and Sections of Convex Bodies	119	(12)
A. Koldobsky
D. Ryabogin
Artem Zvavitch
The Study of Translational Tiling with Fourier Analysis	131	(58)
Mihail N. Kolountzakis
Discrete Maximal Functions and Ergodic Theorems Related to Polynomials	189	(20)
Akos Magyar
What Is It Possible to Say About an Asymptotic of the Fourier Transform of the Characteristic Function of a Two-dimensional Convex Body with Nonsmooth Boundary?	209	(8)
A.N. Podkorytov
Some Recent Progress on the Restriction Conjecture	217	(28)
Terence Tao
Average Decay of the Fourier Transform	245
Giancarlo Travaglini