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

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 ${\frak F} = \{f_i\}_{i\in I}$ and ${\frak E} = \{e_j\}_{j\in G}$ ($G$ a discrete abelian group), relating the decay of the expansion of the elements of ${\frak F}$ in terms of the elements of ${\frak E}$ via a map $a : I \rightarrow G$. A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of ${\frak F}$, the relative measure of ${\frak E}$ — 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.