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.
The principle investigator worked on problems in the structure theory of Banach spaces, more specifically problems which involved certain coordinate systems. During the award period the Principal Investigator supervised two Ph.D. students, Ryan Causey and Keaton Hamm. Ryan Causey is expected to graduate in August, 2014. He wrote two papers on the Szlenk Index of Banach spaces, one will appear in Studia Mathematica, while the second was recently submitted to Fundamenta Mathematica. Keaton Hamm is expected to graduate in August, 2015 and recently succeeded to proof a generalization of the sampling result by Hangelbrook, Madych, Narcowich, and Ward (Journal of Fourier Analysis and Applications, 2012) to the case in which the interpolation points are scattered. His results are in the process of being written up. Together with his former student Daniel Freeman, Richard Haydon, and Edward Odell, the PI was able to use a previous embedding result, and prove that every reflexive Banach space embeds into one, on which all operators are a compact perturbation of a scalar multiple of the identity. This result was independently obtained by Argyros, Raikoftsalis and Zisimopoulou. It was therefore decided to publish this result jointly in The Journal of Functional Analysis. The PI continued his investigation of Greedy bases and algorithms. Jointly with Dilworth, Kutzarova and Wojtaszczyk he studied he studied the new notion of "branch greedy algorithms". With Dilworth, Freeman, and Odell he constructed a greedy basis in Bessov space, and recently he was able to prove, together with Dilworth, Kuztarova and Zsak, that every Banach space which has a greedy basis can be renormed to have an almost 2-greedy basis. Together with his Daniel Freeman, Edward Odell and Andras Zsak, he succeeded to proof that a unconditional basic sequence in L_p(R) which results from shifting a single function in L_p(R) cannot generate all of the space L_p(R). This work will appear in the Israel Journal of Mathematics. With Peter Hajek he found the optimal bound for the dentability index of a Banach space in terms of its szlenk index. This work was submitted to the Bulletin of the London Mathematical Society. Recently the principle Investigator discovered a new proof of Zippin's famous Embedding Theorem which states that every reflexive separable Banach space embeds in one with a basis. This new approach, proves among other things, that a Banach space can embedded into a superspace with basis having the having the same Szlenk index , and solves there fore question by Pelczynski.
