  This work developes a quantitative framework for describing the
overcompleteness of a large class of frames. A previous paper introduced
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 $a \colon I
\to G$. This paper 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 paper, a comprehensive
examination of the interrelations among these localization and approximation
concepts is made, with most implications shown to be sharp.
