Redundancy for localized frames

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 l1-localized frames. We then specialize our results to Gabor multi-frames with generators in M1(Rd), and Gabor molecules with envelopes in W(C, l1). As a main tool in this work, we show there is a universal function g(x) so that, for every e =s> 0, every Parseval frame {fi}i=1M for an N-dimensional Hilbert space HN has a subset of fewer than (1+e)N elements which is a frame for HN with lower frame bound g(e/(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].