Banach Spaces: Theory and Applications

巴拿赫空间的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。本提案的主题是Banach空间的某些“坐标系统”的研究，例如基、框架和字典。所考虑的问题之一是调和分析中的一个老问题，即实线上的p可积函数空间是否具有由同一元素的平移形成的基。我们打算用巴拿赫空间理论中的工具来解决这个问题。另一个突出的问题是将Banach空间嵌入到具有附加结构的Banach空间中，例如嵌入到具有很少算子的Banach空间中。首席研究员打算在这里使用他与E. Odell合作开发的无限渐近对策方法。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，是是否存在算子很少的自反空间的问题。首席研究员打算研究巴拿赫空间结构理论中长期存在的问题，其中许多问题源于或与其他数学领域相关，如信号处理和数据压缩等应用领域，或理论领域，如集合论、谐波分析和近似理论。所采用的技术将涉及分析、几何、无限组合学和逻辑的结合。拟议的工作也可促进后两个领域的进一步发展。该建议的几个部分处理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）向量可以用很少的非零坐标的向量近似，使用易于实现的算法，使表示满足某些稳定性条件。研究者最近对带限函数的研究为凸几何和谐波分析之间的界面研究提供了一条有趣且可能富有成果的研究路线。

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.