Discrete Gabor frames in $\ell ^2(\mathbb {Z}^d)$
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- by Jerry Lopez and Deguang Han PDF
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Abstract:
The theory of Gabor frames for the infinite dimensional signal/ function space $L^{2}(\mathbb {R}^d)$ and for the finite dimensional signal space $\mathbb {R}^{d}$ (or $\mathbb {C}^{d}$) has been extensively investigated in the last twenty years. However, very little has been done for the Gabor theory in the infinite dimensional discrete signal space $\ell ^2(\mathbb {Z}^d)$, especially when $d > 1$. In this paper we investigate the general theory for discrete Gabor frames in $\ell ^2(\mathbb {Z}^d)$. We focus on a few fundamental aspects of the theory such as the density/incompleteness theorem for frames and super-frames, the characterizations for dual frame pairs and orthogonal (strongly disjoint) frames, and the existence theorem for the tight dual frame of the Gabor type. The existence result for Gabor frames (resp. super-frames) requires a generalization of a standard result on common subgroup coset representatives.References
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Additional Information
- Jerry Lopez
- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- Email: jerry@abstractnext.com
- Deguang Han
- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- Email: deguang.han@ucf.edu
- Received by editor(s): May 23, 2011
- Received by editor(s) in revised form: January 11, 2012
- Published electronically: July 15, 2013
- Additional Notes: Part of this paper was reported in the first author’s Ph.D. dissertation, “Optimal Dual Frames for Erasures and Discrete Gabor Frames”, University of Central Florida, 2009.
The second author was supported by NSF grant DMS-1106934. - Communicated by: Thomas Schlumprecht
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3839-3851
- MSC (2010): Primary 42C15, 46C05, 47B10
- DOI: https://doi.org/10.1090/S0002-9939-2013-11875-7
- MathSciNet review: 3091773