Banach Spaces: Theory and Applications

巴拿赫空间，其几何和拓扑结构，为研究动力系统，微分方程，物理学和信号分析提供了一个自然的框架。 这个建议的一个主要主题是调查某些“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，它要求是否空间的p-可积函数的真实的线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。另一个突出的问题是嵌入的Banach空间到Banach空间的一个额外的结构，例如到Banach空间与很少的运营商。主要研究人员打算在这里承担无限渐近游戏的方法，他与E。奥德尔。一个特别重要的问题，特别是在著名的不变子空间问题的背景下，是是否存在具有很少算子的自反空间的问题。主要研究人员打算在长期存在的问题上的结构理论的Banach空间，其中许多人要么起源于，或与其他领域的数学，如应用领域，如信号处理和数据压缩或理论领域，如集合论，调和分析和逼近理论。所采用的技术将涉及分析，几何，无限组合学和逻辑的组合。拟议的工作还可以促进后几个领域的进一步发展。 本提案的几个部分涉及源自信号处理和数据压缩的问题。在这里，人们寻找基、框架，或者更一般的空间字典，其中（某些）向量可以用几个非零坐标的向量来近似，使用容易实现的算法，使得表示满足某些稳定性条件。最近的工作频带有限的职能调查产生了一个有趣的和潜在的富有成效的调查线之间的接口凸几何和调和分析。

Banach spaces, their geometric and topological structure, provide a natural framework for studying dynamical systems, differential equations, physics, and signal analysis. 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 p-integrable functions on the real line 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 problem is the embedding of Banach spaces into Banach spaces with an additional structure, for example into Banach spaces with very few operators. The principal investigator intends to bring to bear here the method of infinite asymptotic games, which he developed in collaboration with E. Odell. A particularly important problem, especially in the context of the famous invariant subspace problem, is the question whether there are reflexive spaces with very few operators. The principal investigator intends to work on longstanding problems in the structure theory of Banach spaces, many of them either originating from, or being related to other area of mathematics such as applied areas like signal processing and data compression or rather theroretical areas like set theory, harmonic analysis, and approximation theory. The techniques to be employed will involve a combination of analysis, geometry, infinite combinatorics, and logic. The proposed work could also spur further development in these latter areas. Several parts of this proposal deal with issues originating in 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 non zero coordinates, using easily implementable algorithms, so that the representation satisfies certain stability conditions. Recent work on band limited functions by the investigator generated an interesting and potentially fruitful line of investigation in the interface between convex geometry and harmonic analysis.