Title

The Uniqueness Of The Dual Of Weyl-Heisenberg Subspace Frames

Keywords

Dual frame; Group-like unitary systems; Projective unitary representations; Von Neumann algebras; Weyl-Heisenberg (Gabor) frame; Zak transform

Abstract

From the Weyl-Heisenberg (WH) density theorem, it follows that a WH-frame (gmα,nβ)m,n∈Z for L2(R) has a unique WH-dual if and only if αβ=1. However, the same argument does not apply to the subspace WH-frame case and it is not clear how to use standard methods of Fourier analysis to deal with this situation. In this paper, we apply operator algebra theory to obtain a very simple necessary and sufficient condition for a given frame (induced by a projective unitary representation of a discrete group) to admit a unique dual (induced by the same system). As a special case, we obtain a characterization for the subspace WH-frames that have unique WH-duals (within the subspace). Using this characterization and the Zak transform, we are able to prove that if (gmα,nβ) m,n∈Z is a WH-frame for a subspace M of L2(ℝ), then, (i) (gmα,nβ)m,n∈ℤ has a unique WH-dual in M when αβ is an integer; (ii) if αβ is irrational, then (gmα,nβ)m,n∈ℤ has a unique WH-dual in M if and only if (gmα,nβ) m,n∈ℤ is a Riesz sequence; (iii) if αβ<1, then the WH-dual for (gmα,nβ)m,n∈Z in M is not unique. © 2004 Elsevier Inc. All rights reserved.

Publication Date

1-1-2004

Publication Title

Applied and Computational Harmonic Analysis

Volume

17

Issue

2 SPEC. ISS.

Number of Pages

226-240

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.acha.2004.04.001

Socpus ID

4344606411 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/4344606411

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