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Discrete Gabor frames in $ \ell^2(\mathbb{Z}^d)$


Authors: Jerry Lopez and Deguang Han
Journal: Proc. Amer. Math. Soc. 141 (2013), 3839-3851
MSC (2010): Primary 42C15, 46C05, 47B10
Published electronically: July 15, 2013
MathSciNet review: 3091773
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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.


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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

DOI: https://doi.org/10.1090/S0002-9939-2013-11875-7
Keywords: Frames, discrete Gabor frames, Weyl-Heisenberg frames
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.