Deformation of Gabor systems

Abstract

We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurling's characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames "without inequalities" from lattices to non-uniform sets.Comment: 31 pages, 2 figure

Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions

“…The technical novelty is a Wiener-type lemma for infinite matrices with off-diagonal decay that are left-invertible on a subspace (Lemma 7.1). This clarifies some subtleties in [1] and [14], and enhances their applicability (in [14] we used so-called Wilson bases to reach similar conclusions).…”

“…So far, strict density conditions have been proved only in few situations, namely (i) for weighted Fock spaces A p φ in dimension 1 [19], and (ii) for Gabor frames, where the result is known as the Balian-Low theorem. Although there is an extensive literature on the theorem for Gabor frames over a lattice, strict density conditions for non-uniform Gabor frames were shown only recently in [2] (with pseudodifferential operators) and [14].…”

“…We note that the proofs in [19] for weighted Fock spaces in one variable rely on the sufficiency of density conditions for sampling and interpolation [6], and these are not available in higher dimension. We will adopt a different approach that combines the strategies of [19] and of [14]. To circumvent arguments that are specific to one-dimensional complex analysis, we resort instead to techniques from [14] that were introduced originally to study the stability of Gabor frames under quite general deformations.…”

“…We will adopt a different approach that combines the strategies of [19] and of [14]. To circumvent arguments that are specific to one-dimensional complex analysis, we resort instead to techniques from [14] that were introduced originally to study the stability of Gabor frames under quite general deformations. More precisely, we consider the notion of Lipschitz convergence of sets.…”

“…See Section 6.1 for details. For the proof of the main results we will enrich the outline of [19] and [14] by several new aspects:…”