Shift-invariant spaces and Gabor systems are frequently used in approximation theory and signal processing. In these settings, it is advantageous for the generators of such spaces to be localized and for the spaces to be representative of a large class of functions. For Gabor systems, the celebrated Balian-Low Theorem shows that if the integer translations and modulations of a function in $L^2(R)$ form an orthonormal basis for $L^2(R)$, then either the function or its fourier transform must be poorly localized. The present work shows that similar results hold in certain shift-invariant spaces. In particular, if the integer translates of a well-localized function in $L^2(R^d)$ form a frame for the shift-invariant space, $V$, generated by the function then $V$ cannot be invariant under any non-integer shift. Similar results are proven under a variety of different basis properties, for finitely many generating functions, and with the extra-invariance property replaced with redundancy in the translates of the generator. Examples are given showing the sharpness of these results.
| Identifer | oai:union.ndltd.org:VANDERBILT/oai:VANDERBILTETD:etd-03252016-124257 |
| Date | 09 April 2016 |
| Creators | Northington V, Michael Carr |
| Contributors | Alexander Powell, Doug Hardin, Akram Aldroubi, Gieri Simonett, Yevgeniy Vorobeychik |
| Publisher | VANDERBILT |
| Source Sets | Vanderbilt University Theses |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.library.vanderbilt.edu/available/etd-03252016-124257/ |
| Rights | unrestricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to Vanderbilt University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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