Density, Overcompleteness, and Localization of Frames. II. Gabor Systems

Abstract

ABSTRACT. This work develops a quantitative framework for describing the overcompleteness of a large class of frames. A previous article introduced notions of localization and approximation between two frames F = {f i } i∈I and E = {e j } j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a : I → G. This article shows that those abstract results yield an array of new implications for irregular Gabor frames. Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane.The notions of localization and related approximation properties are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this article, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with most implications shown to be sharp.

Redundancy for localized frames

“…In the special case where the generator f is a Gaussian function, it has been shown (See [22,24,25] and Gröchenig's book [16]) that the lower density of the set Λ is greater than one if and only if F is a frame for the whole space L 2 (R). This result, combined with results from [1,2], shows that property 4) holds for Gaussian generators. To see this, assume that g is Gaussian, Λ ⊂ R 2 and (g, Λ) generates a Gabor frame for L 2 (R).…”

“…Here, we show that the redundancy function alluded to in [1,2] and [3], for l 1 localized frames, has property 4). We show that for any , every l 1 localized frame with redundancy R has a subframe with redundancy 1 + .…”

“…Choose α, β > 0 so that (αβ) d = 1 − ε 2 . First by Theorem 2.d in [2], it follows that (G(γ, αZ d × βZ d ), i) is a l 1 -selflocalized frame for L 2 (R d ).…”

“…This work, however, left open property 4) for frames with infinite excess (which include, for example, Gabor frames that are not Riesz bases). A quantitative approach to a large class of frames with infinite excess (including Gabor frames) was given in [1,2] which introduced a general notion of a localized frame and, among other results, provided nice quantitative measures associated to this class of frames. From a slightly different angle, the work in [3] quantified overcompleteness for all frames that share a common index set.…”

“…Both papers identify this density function as a good candidate for redundancy, show that the function satisfies properties 1)-3) above and remark that a proper notion of redundancy should also possess property 4). A weak partial result related to property 4) was given in [1,2] where it was shown that for any localized frame F of redundancy R there exists an > 0 and a subframe of F with redundancy R − .…”