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Spectrally Uniform Frames and Spectrally Optimal Dual Frames

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Date Issued:
2013
Abstract/Description:
Frames have been useful in signal transmission due to the built in redundancy. In recent years, theerasure problem in data transmission has been the focus of considerable research in the case theerror estimate is measured by operator (or matrix) norm. Sample results include the characterizationof one-erasure optimal Parseval frames, the connection between two-erasure optimal Parsevalframes and equiangular frames, and some characterization of optimal dual frames.If iterations are allowed in the reconstruction process of the signal vector, then spectral radiusmeasurement 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 optimalframes with respect to the spectral radius measurement) in terms of the redundancy distributionof the frame. Our characterization relies on the connection between spectrally optimal frames andthe linear connectivity property of the frame. We prove that the linear connectivity property isequivalent to the intersection dependence property, and is also closely related to the well-knownconcept of k-independent set. For spectrally two-uniform frames, it is necessary that the framemust be linearly connected. We conjecture that it is also necessary that a two-uniform frame mustbe n-independent. We confirmed this conjecture for the case when N = n+1, n+2, where N is thenumber of vectors in a frame for an n-dimensional Hilbert space. Additionally we also establishseveral necessary and sufficient conditions for the existence of an alternate dual frame to make the iterated reconstruction to work.
Title: Spectrally Uniform Frames and Spectrally Optimal Dual Frames.
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Name(s): Pehlivan, Saliha, Author
Han, Deguang, Committee Chair
Mohapatra, Ram, Committee CoChair
Sun, Qiyu, Committee Member
Tatari, Mehmet, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2013
Publisher: University of Central Florida
Language(s): English
Abstract/Description: Frames have been useful in signal transmission due to the built in redundancy. In recent years, theerasure problem in data transmission has been the focus of considerable research in the case theerror estimate is measured by operator (or matrix) norm. Sample results include the characterizationof one-erasure optimal Parseval frames, the connection between two-erasure optimal Parsevalframes and equiangular frames, and some characterization of optimal dual frames.If iterations are allowed in the reconstruction process of the signal vector, then spectral radiusmeasurement 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 optimalframes with respect to the spectral radius measurement) in terms of the redundancy distributionof the frame. Our characterization relies on the connection between spectrally optimal frames andthe linear connectivity property of the frame. We prove that the linear connectivity property isequivalent to the intersection dependence property, and is also closely related to the well-knownconcept of k-independent set. For spectrally two-uniform frames, it is necessary that the framemust be linearly connected. We conjecture that it is also necessary that a two-uniform frame mustbe n-independent. We confirmed this conjecture for the case when N = n+1, n+2, where N is thenumber of vectors in a frame for an n-dimensional Hilbert space. Additionally we also establishseveral necessary and sufficient conditions for the existence of an alternate dual frame to make the iterated reconstruction to work.
Identifier: CFE0005111 (IID), ucf:50747 (fedora)
Note(s): 2013-05-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): Frames -- optimal dual frames -- spectrally uniform frames -- spectrally optimal dual frames
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0005111
Restrictions on Access: public 2013-11-15
Host Institution: UCF

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