Dilations and Completions for Gabor Systems
Abbreviated Journal Title
J. Fourier Anal. Appl.
Frames; Projective unitary representations; Time-frequency lattices; Gabor frames; Dual frame pair dilation; Von Neumann algebras; Affine; systems; WEYL-HEISENBERG FRAMES; REPRESENTATIONS; Mathematics, Applied
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 of Fourier Analysis and Applications
"Dilations and Completions for Gabor Systems" (2009). Faculty Bibliography 2000s. 1616.