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Frame representations and Parseval duals with applications to Gabor frames


Author: Deguang Han
Journal: Trans. Amer. Math. Soc. 360 (2008), 3307-3326
MSC (2000): Primary 42C15, 46C05, 47B10
DOI: https://doi.org/10.1090/S0002-9947-08-04435-8
Published electronically: January 30, 2008
MathSciNet review: 2379798
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Abstract: Let $ \{x_{n}\}$ be a frame for a Hilbert space $ H$. We investigate the conditions under which there exists a dual frame for $ \{x_{n}\}$ which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether $ \{x_{n}\}$ can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is obtained in terms of the frame excess. For a frame $ \{\pi(g)\xi: g\in G\}$ induced by a projective unitary representation $ \pi$ of a group $ G$, it is possible that $ \{\pi(g)\xi: g\in G\}$ can have a Parseval dual, but does not have a Parseval dual of the same type. The primary aim of this paper is to present a complete characterization for all the projective unitary representations $ \pi$ such that every frame $ \{\pi(g)\xi: g\in G\}$ (with a necessary lower frame bound condition) has a Parseval dual of the same type. As an application of this characterization together with a result about lattice tiling, we prove that every Gabor frame $ {\bf G}(g, \mathcal{L}, \mathcal{K})$ (again with the same necessary lower frame bound condition) has a Parseval dual of the same type if and only if the volume of the fundamental domain of $ \mathcal{L}\times \mathcal{K}$ is less than or equal to $ \frac{1}{2}$.


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Additional Information

Deguang Han
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: dhan@pegasus.cc.ucf.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04435-8
Keywords: Frames, Parseval duals, frame representations, Gabor frames, lattice tiling
Received by editor(s): February 22, 2005
Received by editor(s) in revised form: October 3, 2006
Published electronically: January 30, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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