Fourier Analysis and Convexity

[Elmajbery]

Fourier Analysis and Convexity
Ed. by Luca Brandolini, Leonardo Coizani, Alex losevich, Giancarlo Travaglini
Applied and Numerical Harmonic Analysis
Birkhäuser | 2004 | ISBN: 0817632638 | 280 pages | PDF | 6 MB

This volume is dedicated to Fourier analysis, convex geometry, and related topics. 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.
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

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.

Contents

• Preface
• Lattice Point Problems: Crossroads of Number Theory, Probability Theory and Fourier Analysis
• Totally Geodesic Radon Transform of Lu-Functions on Real Hyperbolic Space
• Fourier Techniques in the Theory of Irregularities of Point Distribution
• Spectral Structure of Sets of Integers
• 100 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 Restriclion Conjecture
• Average Decay of the Fourier Transform

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