  Redundancy is the qualitative property which makes Hilbert space frames so
useful in practice. However, developing a meaningful quantitative notion of
redundancy for infinite frames has proven elusive. Though quantitative
candidates for redundancy exist, the main open problem is whether a frame with
redundancy greater than one contains a subframe with redundancy arbitrarily
close to one. We will answer this question in the affirmative for
$\ell^1$-localized frames. We then specialize our results to Gabor multi-frames
with generators in $M^1(\R^d)$, and Gabor molecules with envelopes in
$W(C,l^1)$. As a main tool in this work, we show there is a universal function
$g(x)$ so that for every $\epsilon>0$, every Parseval frame $\{f_i\}_{i=1}^M$
for an $N$-dimensional Hilbert space $H_N$ has a subset of fewer than
$(1+\epsilon)N$ elements which is a frame for $H_N$ with lower frame bound
$g(\epsilon/(2\frac{M}{N}-1))$. This work provides the first meaningful
quantative notion of redundancy for a large class of infinite frames. In
addition, the results give compelling new evidence in support of a general
definition of reudndancy given in [7].
