Complemented block bases of symmetric bases and spectral tetris fusion frame constructions
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[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This work contains of two parts which are totally unrelated. In the first part we consider techniques surrounding the problem whether a complemented subspace spanned by a block basis of a Banach space with unconditional basis always admits an unconditional basis for its complement. We show several reductions of the problem and provide a new class of block bases of symmetric bases for which the answer is positive. In Banach spaces with symmetric basis we introduce the signature and reduce the problem to a question about averages of constant coefficient block bases. We construct a class of (non-block) subspaces in Banach spaces with symmetric basis and type, which are guaranteed to have an unconditional basis. In the second part we consider implementable constructions for (fusion) frames with prescribed properties based on spectral tetris. We generalize spectral tetris in different ways to construct frames with prescribed sets of eigenvalues an lengths for the frame vectors; working for example with discrete Fourier transform matrices as building blocks. We use these constructions to generate fusion frames with prescribed spectrum for the fusion frame operator and prescribed dimensions for the subspaces, starting in our most general case from a certain reference fusion frame we introduce.
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