Banach Spaces: Theory and Applications
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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.