Algorithm-free methods in fusion frame construction
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Over the past decade numerous papers have been published with novel methods for proving the existence of and constructing fusion frames under various restrictions on the ambient Hilbert space, set of subspace dimensions, weights, and fusion frame operator eigenvalues. A unifying theme in many of these methods is their use of algorithmic, iterative, and recursive constructions -- features which tend to obscure underlying patterns and inhibit deeper analysis. In this thesis we analyze various algorithms prominent in fusion frame construction and, to the greatest extent practicable, derive closed-form expressions describing the frames and fusion frames they generate. Additionally, we thoroughly analyze the process of iterating Naimark and spatial complements and develop explicit, closed-form expressions which illuminate underlying structural relationships and provide a relatively convenient method for classifying and constructing arbitrary tight fusion frames.
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Ph. D.