Banach spaces: Theory and Application

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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.