Density, Overcompleteness, and Localization of Frames. I. Theory

Abstract

ABSTRACT. Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces 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. A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of F , the relative measure of E -both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements -and the density of the set a(I) in G.Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results.The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.Math Subject Classifications. 42C15, 46C99.

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).…”

“…In [3], a redundancy function for infinite frames was defined. It was shown that this redundancy function, when translated to the localized setting dealt with in [1], corresponded to the density measures described in [3]. 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).…”

“…Now, given > 0, applying the techniques from [1] (See the proof of Theorem 8) we can find a subset Σ ⊂ Λ (even of uniform density) so that…”

“…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 − .…”