Dilations and Completions for Gabor Systems

    Authors

    D. G. Han

    Comments

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    Abbreviated Journal Title

    J. Fourier Anal. Appl.

    Keywords

    Frames; Projective unitary representations; Time-frequency lattices; Gabor frames; Dual frame pair dilation; Von Neumann algebras; Affine; systems; WEYL-HEISENBERG FRAMES; REPRESENTATIONS; Mathematics, Applied

    Abstract

    Let Lambda = K x L be a full rank time-frequency lattice in R(d) x R(d). In this note we first prove that any dual Gabor frame pair for a Lambda-shift invariant sub-space M can be dilated to a dual Gabor frame pair for the whole space L(2)(R(d)) when the volume v(Lambda) of the lattice Lambda satisfies the condition v(Lambda) < = 1, and to a dual Gabor Ricsz basis pair for a Lambda-shift invariant subspace containing M when v(Lambda) > 1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7: 419-433, 2001) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel-Gabor family G(g, Lambda) can be completed to a tight Gabor (multi-)frame G(g, Lambda) boolean OR (boolean OR(N)(j=1) G(g(j), Lambda)) for L(2)(R(d)). We show that this is true whenever v(Lambda) < = N. In particular, when v(Lambda) < = 1, any Bessel-Gabor system is a subset of a tight Gabor frame G(g, Lambda) boolean OR G(h, Lambda) for L(2)(R(d)). Related results for affine systems are also discussed.

    Journal Title

    Journal of Fourier Analysis and Applications

    Volume

    15

    Issue/Number

    2

    Publication Date

    1-1-2009

    Document Type

    Article

    Language

    English

    First Page

    201

    Last Page

    217

    WOS Identifier

    WOS:000265398000004

    ISSN

    1069-5869

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