Abstract: | 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 ℓ
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 ε =s> 0, every Parseval frame {f
i
}
i=1
M
for an N-dimensional Hilbert space H
N
has a subset of fewer than (1+ε)N elements which is a frame for H
N
with lower frame bound g(ε/(2M/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 redundancy given in 5]. |