Applications of Fourier Analysis to Banach Space Theory

PI计划利用傅立叶分析的方法研究有限维赋范空间和凸体。其思想是用傅里叶变换表示凸体的不同性质，然后用傅立叶分析方法解决几何问题。这种方法已经得到了几个结果，包括凸体截面上的Busemann-Petty问题的完全解析解。PI计划发展凸体的傅里叶变换、球谐函数和平行截面函数(X射线)之间的新联系，这将导致凸体、交点和投影体的临界截面的新结果，以及Busemann-Petty类型的新结果。这项工作的一个重要部分将与Banach空间局部理论的几个公开问题有关。这包括对切片问题的几种可能的傅立叶分析方法和凸体的中心极限性质的研究。赋范空间几何的一个重要方向是基于凸体的截面和投影的性质来研究凸体的几何性质。这个方向已经在工程和医学上找到了许多应用，从X射线和几何断层成像的经典理论开始。这个项目的目标是发展一种新的方法来研究凸体的截面和投影，基于傅立叶分析的方法。最近PI发现了凸体截面和傅立叶变换之间的新联系，并已经导致了关于较小的物体是否可以具有一致较大的中心截面的Busemann-Petty问题的解析解。与傅立叶分析和几何层析的联系将进一步将Banach空间理论的方法和结果与数学、工程和医学的不同领域联系起来。计算傅里叶变换的新技术将具有独立的价值，也可以应用于信号处理和统计。

The PI plans to study finite dimensional normed spaces and convex bodiesusing methods of Fourier analysis. The idea is to express differentproperties of convex bodies in terms of the Fourier transform and thensolve geometric problems using Fourier analytic methods.This approach has already led to several results including acomplete analytic solution to the Busemann-Petty problem on sectionsof convex bodies. The PI plans to develop new connections betweenthe Fourier transform, spherical harmonics and parallel section functions(X-rays) of convex bodies that will lead to new results on critical sectionsof convex bodies, intersection and projection bodies and to new resultsof Busemann-Petty type. A significant part of the work will be related toseveral open problems of the local theory of Banach spaces. This includesseveral possible Fourier analytic approaches to the slicing problem andthe study of the central limit properties of convex bodies.An important direction of the geometry of normed spaces is the study ofgeometric properties of convex bodies based on the properties of sectionsand projections of these bodies. This direction has already foundnumerous applications to engineering and medicine, starting with theclassical theory of X-rays and geometric tomography. The goal of thisproject is to develop a new approach to the study of sections andprojections of convex bodies based on methods of Fourier analysis.New connections between sections of convex bodies and the Fourier transformhave been recently found by the PI and have already led to an analyticsolution to the Busemann-Petty problem of whether smaller bodies can haveuniformly larger central sections. The connections with Fourier analysisand geometric tomography will further relate methods and results of theBanach space theory to different areas of mathematics, engineering andmedicine. New techniques for computing the Fourier transform will haveindependent value and can also be applied to signal processing and statistics.