Applications of Fourier Analysis to Banach Space Theory

PI计划使用傅里叶分析方法研究有限维赋范空间和凸体。其思想是用傅里叶变换来表达凸体的不同性质，然后用傅里叶分析法来解决几何问题。这种方法已经导致了几个结果，包括恒星体截面体积的傅里叶变换公式和凸体截面的Busemann-Petty问题的完整解析解。重点是发展凸体的傅里叶变换和平行截面函数（x射线）之间的新联系，这将导致关于巴拿赫空间子空间结构的新信息，某些类凸体的几何性质。结果也将应用于正定范数相关函数（勋伯格型定理）及其与稳定和各向同性随机过程的联系的研究。这项研究将在数学和工程的几个领域之间提供新的联系。许多结果将在泛函分析、几何和调和分析的语言中具有等效的公式。赋范空间的子空间的研究与几何层析成像密切相关。经典的问题是通过用不同位置的x射线源获得的固体切片的面积来确定固体的几何性质。傅里叶变换方法将有助于解决医学工程中的这一重要问题。统计学的一个应用是研究有限数据下随机过程的不同参数。一个重要的例子是在只能测量最大振幅的情况下确定信号的主频率。计算傅里叶变换的新技术将作为一个重要的工具，并将在信号处理中有其他应用。

DMS-9820848ABSTRACT 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 the Fourier transform formulae for the volume of sections of star bodies and a complete analytic solution to the Busemann-Petty problem on sections of convex bodies. The emphasis is on developing new connections betweenthe Fourier transform and parallel section functions (X-rays) of convex bodies that will lead to new information about the structure of subspaces of Banach spaces, geometric properties of certain classes of convex bodies. The results will also be applied to the study of positive definite norm dependent functions (theorems of Schoenberg's type) and their connections with stable and isotropic stochastic processes. This research will provide new connections between several areas ofmathematics and engineering. Many of the results will have equivalentformulations in the languages of functional analysis, geometry andharmonic analysis. The study of subspaces of normed spaces is closelyrelated to geometric tomography. The classical problem is to determinecertain geometric properties of a solid by means of the areas of itsslices obtained with differently placed sources of X-rays. The Fouriertransform approach will contribute to this important problem in medicalengineering. An application to statistics is the study of differentparameters of stochastic processes under restricted data. An importantexample is the problem of determining the main frequences of a signalwhen only the maximal amplitude can be measured. New techniques forcalculating the Fourier transform will serve as an important tooland will have other applications to signal processing.