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Dilations and Completions for Gabor Systems

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Abstract

Let \(\Lambda=\mathcal{K}\times\mathcal{L}\) be a full rank time-frequency lattice in ℝd×ℝd. In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L 2(ℝd) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>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,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪ N j=1 G(g j ,Λ)) for L 2(ℝd). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L 2(ℝd). Related results for affine systems are also discussed.

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 38216, USA

    Deguang Han

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  1. Deguang Han
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Correspondence to Deguang Han.

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Communicated by Chris Heil.

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Han, D. Dilations and Completions for Gabor Systems. J Fourier Anal Appl 15, 201–217 (2009). https://doi.org/10.1007/s00041-008-9028-y

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  • DOI: https://doi.org/10.1007/s00041-008-9028-y

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