Topics in Banach Space Theory

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This project will investigate some problems in Banach space theory which are motivated in part by applications to signals processing and data compression. Elements of a given Banach space can be approximated by linear combinations of elements drawn from a Schauder basis or from a redundant system such as a frame or a dictionary. The approximants are typically selected by a greedy algorithm such as the Dual Greedy, X-Greedy , or Thresholding Greedy Algorithms. The project will study the convergence of such algorithms in the norm or the weak topology of the Banach space. An important open problem in this area is to show that the X-Greedy Algorithm converges in Lebesgue spaces. Convergence results for greedy algorithms are connected to geometrical properties of the underlying Banach space such as uniform smoothness or the Kadets-Klee property. They are also connected to ``partial unconditionality'' properties of the underlying system such as quasi-greediness. An important related problem in Banach space theory which will be studied is to show that certain absolute constants (the Elton constants) which arise naturally in the study of partial unconditionality are uniformly bounded. A second set of problems in this project concerns coefficient quantization in Banach spaces. The goal is to replace arbitrary real coefficients (with respect to some underlying system) by coefficients that are selected from a finite ``alphabet'' using some algorithmic procedure. Building on recent results in the case of a Schauder basis, existence of systems with desirable quantization properties will be connected to the geometry of the underlying Banach space. A Banach space is a collection of ``vectors'' which can be added together or multiplied by numbers to form other vectors. There is a concept of ``distance'' between vectors which is analogous to the everyday notion of distance between points in the three-dimensional Banach space which we inhabit. There is a wide range of Banach spaces of importance in both pure and applied mathematics which can be distinguished from each other by ``geometrical'' properties such as ``smoothness'' or ``convexity''. Mathematicians have found that Banach spaces provide the appropriate framework in which to formulate major areas of mathematics such as Harmonic Analysis, Partial Differential Equations, and Functional Analysis. Banach spaces are also used by scientists and engineers to model problems in applied areas such as fluid mechanics, signals processing, image compression, and the pricing of financial derivatives. An individual vector belonging to a given Banach space is usually identified by an infinite string of numbers called ``coefficients'' . The problem of data compression is to select the most significant coefficients and to discard the rest. The problem of analog to digital conversion is to ``quantize'' the selected coefficients by binary numbers in such a way that the quantized vector is a good approximation to the ``target'' vector. The geometry of the underlying Banach space will play an important part in the development of effective algorithms for implementing such procedures. In this project we will investigate these and other problems concerning Banach spaces and their geometrical properties.