Banach Spaces: Theory and Applications

Banach 空间：理论与应用

基本信息

批准号：
1464713
负责人：
Thomas Schlumprecht
金额：
$ 26.44万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-05-15 至 2019-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1464713&HistoricalAwards=false
关键词：
Banach Spaces Theory Applications

项目摘要

Banach spaces together with their geometry provide an important framework for studying problems in physics, signal processing, and the analysis of large data sets. This research project on Banach spaces will have two main directions. The first one is the study of coordinate systems of Banach spaces. If we model a given problem in physics or signal analysis using a certain Banach space we will also need an "appropriate coordinate system" for that space, i.e. we want to represent the elements of this space by a sequence of numbers. What constitutes an "appropriate coordinate system" will of course depend on the specific problem, but generally the goal is to approximate a given element of a Banach space as well as possible with the least amount of coordinates, and to reconstruct the element from the given sequence of coordinates with the smallest possible error and the least amount of effort. Not every Banach space admits coordinate systems which have the minimality properties; one would like a coordinate system to satisfy. Therefore one also needs to find for a given Banach space criteria to embed it into a space with coordinate systems, without losing the topological and geometrical properties of the original space. Metric spaces are often used in Computer Science to model large data sets, and our second objective is to investigate embeddings of metric spaces into Banach spaces, and to obtain on the one hand information about the structure of the metric space, from the property that it embeds in certain Banach spaces, and on the other hand deduce geometric properties of a Banach space, from the property that certain metrics embed or do not embed in it. Many of the problems under study in this project either originate from, or are related to, other areas of mathematics such as descriptive set theory, harmonic analysis, metric geometry, and approximation theory. The techniques to be employed will involve a combination of analysis, geometry, infinite combinatorics, and logic. One of the problems considered is an old one from harmonic analysis, which asks whether the space of p-integrable functions and other function spaces have a Schauder basis formed by translates of only one element. Another prominent problem is the embedding of uniformly convex Banach spaces into such spaces with a basis or a finite dimensional decomposition. Together with his colleague Sivakumar and their joint student, Keaton Hamm, the investigator intends to study the representation and approximation of elements of certain function classes using redundant coordinate systems. The research also pursues a new direction and investigates problems in metric geometry. This work aims to characterize geometric and topological properties of Banach space like reflexivity by the embeddability of certain metric spaces.

Banach空间及其几何形状为研究物理学，信号处理和大型数据集的分析提供了重要的框架。这个关于Banach空间的研究项目将有两个主要方向。第一个是研究Banach空间的坐标系统。如果我们使用某个Banach空间在物理学或信号分析中对给定的问题进行建模，我们还需要为该空间进行“适当的坐标系”，即我们要以一系列数字来表示该空间的元素。当然，什么构成“适当的坐标系”将取决于特定的问题，但总体而言，目标是近似Banach空间的给定元素，并以最少的坐标量近似，并从给定的坐标序列中重建元素，并以最小的可能误差和最少的努力和最少的努力和最少的努力重建元素。并非每个Banach空间都接受具有最小属性的坐标系。一个人希望坐标系统满足。因此，对于给定的Banach空间标准也需要找到将其嵌入带有坐标系的空间中的，而不会失去原始空间的拓扑和几何特性。度量空间通常在计算机科学中用于建模大型数据集，我们的第二个目标是调查公制空间的嵌入到巴拉赫空间中，并一方面获取有关公制空间结构的信息，从嵌入某些Banach空间中的属性，另一方面，从某些Meted embed embed embed emb中，它嵌入了某些BANACH空间，另一方面是从其他手中推导出banach空间的信息。该项目中研究的许多问题源于数学的其他领域，例如描述性集理论，谐波分析，度量几何和近似理论。要使用的技术将涉及分析，几何，无限组合和逻辑的组合。考虑的问题之一是谐波分析中的旧问题，该问题询问P-Contegryable功能和其他功能空间是否仅由一个元素翻译形成Schauder基础。另一个突出的问题是将均匀凸出的Banach空间嵌入具有基础或有限尺寸分解的此类空间。研究人员与他的同事Sivakumar和他们的联合学生Keaton Hamm一起研究使用冗余坐标系统研究某些功能类别元素的表示和近似。这项研究还迈出了新的方向，并研究了度量几何学问题。这项工作旨在通过某些度量空间的嵌入性来表征Banach空间的几何和拓扑特性，例如反射性。