The Geometry of Structured Parseval Frames and Frame Potentials

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dc.contributor.advisorBodmann, Bernhard G.
dc.contributor.committeeMemberCasazza, Peter G.
dc.contributor.committeeMemberLabate, Demetrio
dc.contributor.committeeMemberPaulsen, Vern I.
dc.creatorHaas, John Isaac 1981-
dc.creator.orcid0000-0003-4656-7291
dc.date.accessioned2019-09-19T00:57:24Z
dc.date.available2019-09-19T00:57:24Z
dc.date.createdMay 2015
dc.date.issued2015-05
dc.date.submittedMay 2015
dc.date.updated2019-09-19T00:57:25Z
dc.description.abstractIn this dissertation, we study the geometric character of structured Parseval frames, which are families of vectors that provide perfect Hilbert space reconstruction. Equiangular Parseval frames (EPFs) satisfy that the magnitudes of the pairwise inner products between frame vectors are constant. These types of frames are useful in many applications. However, EPFs do not always exist and constructing them is often difficult. To address this problem, we consider two generalizations of EPFs, equidistributed frames and Grassmannian equal-norm Parseval frames, which include EPFs when they exist. We provide several examples of each type of Parseval frame. To characterize and locate these classes of frames, we develop an optimization program involving families of real analytic frame potentials, which are real-valued functions of frames. With the help of the {\L}ojasiewicz gradient inequality, we prove that the gradient descent of these functions on the manifold of Gram matrices of Parseval frames always converges to critical points. We then show that, under certain conditions, the frames corresponding to the Gram matrices of the critical points for different frame potentials possess desirable geometric properties. These properties include the equal-norm, equiangular, non-orthodecomposable, equidistributed and Grassmannian equal-norm cases. We also discuss the history of EPFs and frame potentials and provide a new characterization of EPFs in terms of the Fourier transform. Using this characterization, we reprove a known result regarding cyclic EPFs and difference sets.
dc.description.departmentMathematics, Department of
dc.format.digitalOriginborn digital
dc.format.mimetypeapplication/pdf
dc.identifier.citationPortions of this document appear in: Bodmann, Bernhard G., and John Haas. "Frame potentials and the geometry of frames." Journal of Fourier Analysis and Applications 21, no. 6 (2015): 1344-1383.
dc.identifier.urihttps://hdl.handle.net/10657/4871
dc.language.isoeng
dc.rightsThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).
dc.subjectFrame theory
dc.subjectParseval
dc.subjectEquiangular
dc.subjectEquidistributed
dc.subjectGrassmannian
dc.subjectFrame potentials
dc.subjectGradient descent
dc.titleThe Geometry of Structured Parseval Frames and Frame Potentials
dc.type.dcmiText
dc.type.genreThesis
thesis.degree.collegeCollege of Natural Sciences and Mathematics
thesis.degree.departmentMathematics, Department of
thesis.degree.disciplineMathematics
thesis.degree.grantorUniversity of Houston
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy

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