Density, overcompleteness, and localization of frames

This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames $\mathcal {F} = \{f_i\}_{i \in I}$ and $\mathcal {E} = \{e_j\}_{j \in G}$ ($G$ a discrete abelian group), relating the decay of the expansion of the elements of $\mathcal {F}$ in terms of the elements of $\mathcal {E}$ via a map $a \colon I \to G$. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of $\mathcal {F}$, the relative measure of $\mathcal {E}$—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. These abstract results yield an array of new implications for irregular Gabor frames. Various Nyquist density results for Gabor frames are recovered as special cases, but 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.