Banach Spaces: Theory and Applications

Banach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。这项建议的一个主要主题是研究Banach空间的某些“坐标系”，例如基、框架和词典。所考虑的问题之一是调和分析中的一个老问题，即实直线上的p-可积函数空间是否有由相同元素的平移形成的基。本文试图用Banach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。这位首席研究员打算在这里运用他与E·奥德尔合作开发的无限渐近对策的方法。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，是不是存在算子很少的自反空间的问题。这位首席研究员打算研究Banach空间结构理论中的长期存在的问题，其中许多问题要么源于其他数学领域，要么与其他数学领域相关，例如应用领域如信号处理和数据压缩，或者更确切地说，理论领域如集合论、调和分析和逼近理论。所使用的技术将涉及分析、几何、无限组合和逻辑的组合。拟议的工作还可以促进后两个领域的进一步发展。该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)向量可以用具有很少非零坐标的向量来逼近，使用容易实现的算法，以便表示满足一定的稳定性条件。这位研究人员最近在带限函数方面的工作在凸几何和调和分析之间的界面上产生了一条有趣的、可能富有成效的研究路线。

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.