The Complete Structure of Linear and Nonlinear Deformations of Frames on a Hilbert Space |
Posted on:2017-01-14 | Degree:M.S | Type:Thesis |
University:East Tennessee State University | Candidate:Agrawal, Devanshu | Full Text:PDF |
GTID:2460390014964264 | Subject:Mathematics |
Abstract/Summary: | |
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. |
Keywords/Search Tags: | Hilbert space, Frames, Linear, Gabor frame, Window function |
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