Title

The Existence Of Tight Gabor Duals For Gabor Frames And Subspace Gabor Frames

Keywords

Frame representations; Frames; Gabor frames; Lattice tiling; Parseval duals; Pseudo-duals; Subspace Gabor frame

Abstract

Let K and L be two full-rank lattices in Rd. We give a complete characterization for all the Gabor frames that admit tight dual of the same type. The characterization is given in terms of the center-valued trace of the von Neumann algebra generated by the left regular projective unitary representations associated with the time-frequency lattice K × L. Two applications of this characterization were obtained: (i) We are able to prove that every Gabor frame has a tight dual if and only if the volume of K × L is less than or equal to frac(1, 2). (ii) We are able to obtain sufficient or necessary conditions for the existence of tight Gabor pseudo-duals for subspace Gabor frames in various cases. In particular, we prove that every subspace Gabor frame has a tight Gabor pseudo-dual if either the volume v (K × L) ≤ frac(1, 2) or v (K × L) ≥ 2. Moreover, if K = α Zd, L = β Zd with α β = 1, then a subspace Gabor frame G (g, L, K) has a tight Gabor pseudo-dual only when G (g, L, K) itself is already tight. © 2008 Elsevier Inc. All rights reserved.

Publication Date

1-1-2009

Publication Title

Journal of Functional Analysis

Volume

256

Issue

1

Number of Pages

129-148

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.jfa.2008.10.015

Socpus ID

55649092269 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/55649092269

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