Banach spaces: Theory and Application

PI将研究巴拿赫空间结构理论中长期存在的问题，其中许多问题起源于或与其他数学领域有关，如集合论、谐波分析和近似理论。该提案的主题是Banach空间的某些“坐标系统”的研究，例如基、框架和字典。其中考虑的一个问题是调和分析中的一个老问题，即平方可积函数的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关于p可积函数空间的（补）子空间的结构。PI打算在这里采用他与E. Odell合作开发的无限渐近对策方法。该建议的几个部分处理来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）向量可以由具有少量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，并且/或可以“量化”。巴拿赫空间的几何和拓扑结构为研究动力系统、微分方程、多分辨率分析提供了一个自然的框架，特别是当人们想要模拟复杂和高维结构时。本提案的主要目标是研究这些空间上的几种类型的坐标系。所采用的技术将包括分析、无限组合学和逻辑的结合。拟议的工作还可以促进这些领域的进一步发展。

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.