Frames, optimal dual frames, spectrally uniform frames, spectrally optimal dual frames
Frames have been useful in signal transmission due to the built in redundancy. In recent years, the erasure problem in data transmission has been the focus of considerable research in the case the error estimate is measured by operator (or matrix) norm. Sample results include the characterization of one-erasure optimal Parseval frames, the connection between two-erasure optimal Parseval frames and equiangular frames, and some characterization of optimal dual frames. If iterations are allowed in the reconstruction process of the signal vector, then spectral radius measurement for the error operators is more appropriate then the operator norm measurement. We obtain a complete characterization of spectrally one-uniform frames (i.e., one-erasure optimal frames with respect to the spectral radius measurement) in terms of the redundancy distribution of the frame. Our characterization relies on the connection between spectrally optimal frames and the linear connectivity property of the frame. We prove that the linear connectivity property is equivalent to the intersection dependence property, and is also closely related to the well-known concept of k-independent set. For spectrally two-uniform frames, it is necessary that the frame must be linearly connected. We conjecture that it is also necessary that a two-uniform frame must be n-independent. We confirmed this conjecture for the case when N = n+1, n+2, where N is the number of vectors in a frame for an n-dimensional Hilbert space. Additionally we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iii iterated reconstruction to work.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Dissertations, Academic -- Sciences, Sciences -- Dissertations, Academic
Pehlivan, Saliha, "Spectrally Uniform Frames And Spectrally Optimal Dual Frames" (2013). Electronic Theses and Dissertations. 2724.