Degree Name

MS (Master of Science)

Program

Mathematical Sciences

Date of Award

5-2016

Committee Chair or Co-Chairs

Jeff Knisley

Committee Members

Anant Godbole, Michelle Joyner, Debra Knisley

Abstract

A frame is a possibly linearly dependent set of vectors in a Hilbert space that facilitates the decomposition and reconstruction of vectors. A Parseval frame is a frame that acts as its own dual frame. A Gabor frame comprises all translations and phase modulations of an appropriate window function. We show that the space of all frames on a Hilbert space indexed by a common measure space can be fibrated into orbits under the action of invertible linear deformations and that any maximal set of unitarily inequivalent Parseval frames is a complete set of representatives of the orbits. We show that all such frames are connected by transformations that are linear in the larger Hilbert space of square-integrable functions on the indexing space. We apply our results to frames on finite-dimensional Hilbert spaces and to the discretization of the Gabor frame with a band-limited window function.

Document Type

Thesis - Open Access

Copyright

Copyright by the authors.

Included in

Analysis Commons

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