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.
Identifer | oai:union.ndltd.org:ETSU/oai:dc.etsu.edu:etd-4381 |
Date | 01 May 2016 |
Creators | Agrawal, Devanshu |
Publisher | Digital Commons @ East Tennessee State University |
Source Sets | East Tennessee State University |
Language | English |
Detected Language | English |
Type | text |
Format | application/pdf |
Source | Electronic Theses and Dissertations |
Rights | Copyright by the authors. |
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