Applications of Fourier Analysis to Banach Space Theory

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

The PI plans to study finite dimensional normed spaces and convex bodies using methods of Fourier analysis. The idea is to express different properties of convex bodies in terms of the Fourier transform and then solve geometric problems using Fourier analytic methods. This approach has already led to several results including a complete analytic solution to the Busemann-Petty problem on sections of convex bodies. The PI plans to develop new connections between the Fourier transform, spherical harmonics and parallel section functions (X-rays) of convex bodies that will lead to new results on critical sections of convex bodies, intersection and projection bodies and to new results of Busemann-Petty type. A significant part of the work will be related to several open problems of the local theory of Banach spaces. This includes several possible Fourier analytic approaches to the slicing problem and the study of the central limit properties of convex bodies. An important direction of the geometry of normed spaces is the study of geometric properties of convex bodies based on the properties of sections and projections of these bodies. This direction has already found numerous applications to engineering and medicine, starting with the classical theory of X-rays and geometric tomography. The goal of this project is to develop a new approach to the study of sections and projections of convex bodies based on methods of Fourier analysis. New connections between sections of convex bodies and the Fourier transform have been recently found by the PI and have already led to an analytic solution to the Busemann-Petty problem of whether smaller bodies can have uniformly larger central sections. The connections with Fourier analysis and geometric tomography will further relate methods and results of the Banach space theory to different areas of mathematics, engineering and medicine. New techniques for computing the Fourier transform will have independent value and can also be applied to signal processing and statistics.