Applications of Fourier Analysis to Banach Space Theory

PI计划使用傅立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示凸体的不同性质，然后用傅里叶分析方法来解决几何问题。这种方法已经导致了几个结果，包括傅立叶变换公式的体积部分的星星机构和一个完整的解析解的Busemann-Petty问题的部分凸体。重点是发展新的连接betweenthe傅立叶变换和平行截面函数（X射线）的凸体，这将导致新的信息结构的子空间的Banach空间，几何性质的某些类别的凸体。结果也将被应用到正定范数依赖函数（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。这项研究将为数学和工程学的几个领域提供新的联系。许多结果在泛函分析、几何和调和分析的语言中具有等价的公式。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射线源获得的切片面积来确定固体的某些几何性质。傅立叶变换方法将有助于解决医学工程中的这一重要问题。在统计学中的一个应用是在有限数据下研究随机过程的不同参数。一个重要的例子是当只能测量信号的最大幅度时，如何确定信号的主要频率。计算傅里叶变换的新技术将作为一种重要的工具，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.