Frame representations and Parseval duals with applications to Gabor frames
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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 $\textbf {G}(g, \mathcal {L}, \mathcal {K})$ (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 $\frac {1}{2}$.References
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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
- Received by editor(s): February 22, 2005
- Received by editor(s) in revised form: October 3, 2006
- Published electronically: January 30, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2379798