Dilations and Completions for Gabor Systems

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.