Banach spaces: Theory and Application

PI将致力于解决Banach空间结构理论中的长期问题，其中许多问题要么起源于其他数学领域，要么与其他数学领域有关，如集合论、调和分析和逼近理论。这一建议的一个主要主题是调查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，它询问平方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。另一个突出的问题涉及p-可积函数空间的(补)子空间的结构。这位少年派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。该提案的几个部分涉及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)向量可以用具有很少非零坐标的向量来逼近，使用容易实现的算法，从而使表示满足某些稳定性条件，和/或可以被‘量化’。Banach空间，它们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，特别是当一个人想要模拟复杂和高维结构时。这项提议的一个主要目标是研究这些空间上的几种类型的坐标系。所使用的技术将涉及分析、无限组合和逻辑的组合。拟议的工作还可能刺激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.