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.

巴纳赫空间及其几何结构为研究物理、信号处理和大数据集分析问题提供了重要的框架。这个巴拿赫空间的研究项目将有两个主要方向。第一个是Banach空间坐标系的研究。如果我们使用某个巴拿赫空间对物理或信号分析中的给定问题进行建模，我们还需要该空间的“适当的坐标系”，即我们希望用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”取决于具体问题，但通常目标是用最少的坐标尽可能地逼近 Banach 空间的给定元素，并根据给定的元素重建该元素具有最小可能误差和最少工作量的坐标序列。并非每个 Banach 空间都承认具有极小性的坐标系；人们想要一个坐标系来满足。因此，我们还需要找到给定的Banach空间准则，将其嵌入到具有坐标系的空间中，而不丢失原始空间的拓扑和几何性质。度量空间在计算机科学中经常用于对大型数据集进行建模，我们的第二个目标是研究度量空间到 Banach 空间的嵌入，一方面从度量空间的属性中获取有关度量空间结构的信息嵌入到某些 Banach 空间中，另一方面，根据某些度量嵌入或不嵌入其中的属性推导出 Banach 空间的几何属性。该项目研究的许多问题要么源于数学的其他领域，要么与之相关，例如描述性集合论、调和分析、度量几何和逼近论。所采用的技术将涉及分析、几何、无限组合和逻辑的组合。考虑的问题之一是调和分析中的一个老问题，它询问 p 可积函数的空间和其他函数空间是否具有仅由一个元素平移形成的 Schauder 基。另一个突出的问题是将一致凸巴拿赫空间嵌入到具有基或有限维分解的空间中。研究人员打算与他的同事 Sivakumar 和联合学生 Keaton Hamm 一起研究使用冗余坐标系的某些函数类元素的表示和近似。该研究还追求一个新的方向并研究度量几何中的问题。这项工作旨在通过某些度量空间的可嵌入性来表征巴拿赫空间的几何和拓扑性质，例如自反性。