Banach spaces: Theory and Application

巴纳赫空间：理论与应用

• 批准号: 0856148
• 负责人: Thomas Schlumprecht
• 金额: $ 25.23万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0856148&HistoricalAwards=false
• 关键词: Banach; spaces; Theory; Application

The PI will work on longstanding problems in the structure theory of Banach spaces, many of them either originating or being related to other areas of mathematics such as set theory, harmonic analysis and approximation theory. A main theme of this proposal is the investigation of certain ``coordinate systems'' for Banach spaces, e.g. bases, frames, and dictionaries. One of the problems considered is an old one from harmonic analysis, which asks whether the space of square integrable functions has a basis formed by translates of the same element. It is intended to attack this problem with tools from the theory of Banach spaces. Another prominent question deals with the structure of the (complemented) subspaces of the space of p-integrable functions. The PI intends to bring to bear here the method of infinite asymptotic games, which he has developed in collaboration with E. Odell. Several parts of this proposal deal with other issues originating from signal processing and data compression. Here one looks for bases, frames, or, more generally dictionaries of spaces, in which (certain) vectors can be approximated by vectors with few nonzero coordinates, using easily implementable algorithms, so that the representation satisfies certain stability conditions, and/or can be ``quantized''.Banach spaces, their geometric and topological structure provide a natural framework for studying dynamical systems, differential equations, multi-resolution analysis, in particular if one wants to model complex and high-dimensional structures. A main goal of this proposal is to study several types of coordinate systems on these spaces. The techniques to be employed will involve a combination of analysis, infinite combinatorics, and logic. The proposed work could also spur further development in these areas.

PI将致力于解决Banach空间结构理论中的长期问题，其中许多问题要么源于数学的其他领域，如集合论，调和分析和逼近理论。这个建议的一个主题是调查某些"坐标系“的Banach空间，例如基地，框架和字典。考虑的问题之一是一个旧的从调和分析，它问是否空间的平方可积函数有一个基础所形成的平移相同的元素。它的目的是攻击这个问题的工具从理论的Banach空间。 另一个突出的问题涉及p-可积函数空间的（补）子空间的结构。PI打算在这里承担无限渐近游戏的方法，他与E。奥德尔。 本提案的几个部分涉及源自信号处理和数据压缩的其他问题。在这里，人们寻找基，框架，或者更一般的空间字典，其中（某些）向量可以近似为具有很少非零坐标的向量，使用易于实现的算法，使得表示满足某些稳定性条件，和/或可以“量化”。Banach空间，它们的几何和拓扑结构为研究动力系统，微分方程，多分辨率分析，特别是如果想要对复杂和高维结构进行建模。这个建议的一个主要目标是研究这些空间上的几种类型的坐标系。所采用的技术将涉及分析，无限组合学和逻辑的组合。拟议的工作还可以促进这些领域的进一步发展。