Banach Spaces: Theory and Application

Banach 空间：理论与应用

• 批准号: 0556013
• 负责人: Thomas Schlumprecht
• 金额: $ 10.96万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0556013&HistoricalAwards=false
• 关键词: Banach; Spaces; Theory; Application

The author will work on problems in Frame Theory and on several fundamental questions in Geometrical Functional Analysis. Frames represent a theoretical underpinning of Signal processing. In the context of A/D conversion it is crucial to quantize signals, or, abstractly speaking, the coefficients in a frame-expansion. Using infinite dimensional Banach space theory the author intends to formulate and prove results which provide the theoretical background for such quantizations. He will build on results already proven for basis-expansions. The author intends also to continue his research on the following fundamental questions of Geometrical Functional Analysis: Does there exist a bounded linear operator on any infinite dimensional Banach space which is not a compact perturbation of a multiple of the identity? The principal investigator proposes to explore several interesting variations on this theme, each with a distinctly structure-theoretical flavor. Secondly, the author proposes to work on several structure theoretical questions in Banach spaces, which concern their co-ordinate systems. These problems can be summarized under the following general question: Given a separable Banach space, how close can we find an other space containing the first one which has a finite dimensional decomposition (or even a basis)?

作者将研究框架理论中的问题和几何泛函分析中的几个基本问​​题。框架代表了信号处理的理论基础。在 A/D 转换的背景下，量化信号（或者抽象地说，帧扩展中的系数）至关重要。作者打算使用无限维巴纳赫空间理论来公式化并证明结果，为此类量化提供理论背景。他将在已经证明可以扩大基础的成果的基础上再接再厉。作者还打算继续研究以下几何泛函分析的基本问题：任意无限维Banach空间上是否存在不是单位倍数的紧摄动的有界线性算子？首席研究员建议探索这个主题的几个有趣的变体，每个变体都具有明显的结构理论风格。其次，作者建议研究巴纳赫空间中涉及坐标系的几个结构理论问题。这些问题可以概括为以下一般性问题：给定一个可分离的 Banach 空间，我们能找到一个包含第一个具有有限维分解（甚至是基）的空间的距离有多近？