In this talk I will present a quantitative framework for describing the overcompleteness (redundancy) of a large class of frames. I will introduce the notion of localization between two frames F = {f_i}_{i \in I} and E= {e_j}_{j \in Z^d}, relating the decay of the expansion of the elements of F in terms of the elements of E via a map a:I->Z^d. A fundamental set of equalities are shown between two seemingly unrelated quantities: the relative measure, which is determined by certain averages of <f_i><\tilde{f}_i> (inner products of frame elements with their corresponding canonical dual frame elements, and the density of the set a(I) in Z^d. They read:

p-lim 1/|I_N| \sum_{i\in I_N} <f_i><\tilde{f}_i> = 1/D(a;p)

The idea of localization and the above equalities lead to an array of new results that hold in a general setting, and are novel when applied to the cases of irregular Gabor frames and windowed exponentials. Various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their meaning and implications are clarified.

This is a joint work with Chris Heil, Pete Casazza, and Zeph Landau.