The uniqueness of the dual of Weyl-Heisenberg subspace frames

    Authors

    J. P. Gabardo;D. G. Han

    Comments

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    Abbreviated Journal Title

    Appl. Comput. Harmon. Anal.

    Keywords

    Weyl-Heisenberg (Gabor) frame; Dual frame; Zak transform; projective; unitary representations; group-like unitary systems; von Neumann; algebras; Mathematics, Applied; Physics, Mathematical

    Abstract

    From the Weyl-Heisenberg (WH) density theorem, it follows that a WH-frame (g(malpha,nbeta))(m,n is an element of Z) for L-2(R) has a unique WH-dual if and only if alphabeta = 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 (g(malpha,nbeta))(m,nis an element ofZ) is a W-H-frame for a subspace M of L-2(R), then, (i) (g(malpha,nbeta))(m,nis an element ofZ) has a unique WH-dual in M when alphabeta is an integer; (ii) if alphabeta is irrational, then (g(malpha,nbeta))(m,nis an element ofZ) has a unique WH-dual in M if and only if (g(malpha,nbeta))(m,nis an element ofZ) is a Riesz sequence; (iii) if alphabeta < 1, then the WH-dual for (g(malpha,nbeta))(m,nis an element ofZ) in M is not unique. (C) 2004 Elsevier Inc. All rights reserved.

    Journal Title

    Applied and Computational Harmonic Analysis

    Volume

    17

    Issue/Number

    2

    Publication Date

    1-1-2004

    Document Type

    Article

    Language

    English

    First Page

    226

    Last Page

    240

    WOS Identifier

    WOS:000223869100006

    ISSN

    1063-5203

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