Synopses & Reviews
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
Synopsis
Explores relationship between Fourier Analysis, convex geometry, and related areas; in the past, study of this relationship has led to important mathematical advances Presents new results and applications to diverse fields such as geometry, number theory, and analysis Contributors are leading experts in their respective fields Will be of interest to both pure and applied mathematicians
Table of Contents
Preface
Contributors
Lattice Point Problems: Crossroads of Number Theory, Probability Theory, and Fourier Analysis
Totally Geodesic Radon Transform of L^P-Functions on Real Hyperbolic Space
Fourier Techniques in the Theory of Irregularities of Point Distributions
Spectral Structure of Sets of Integers
One-Hundred Years of Fourier Series and Spherical Harmonics in Convexity
Fourier Analytic Methods in the Study of Projections and Sections of Convex Bodies
The Study of Translational Tiling with Fourier Analysis
Discrete Maximal Functions and Ergodic Theorems Related to Polynomials
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?
Some Recent Progress on the Restriction Conjecture
Average Decay of the Fourier Transform
Index