Redundancy for localized and Gabor 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 ${\ell }^1$-localized frames. We then specialize our results to Gabor multi-frames with generators in $M^1({\mathbb{R}}^d)$, 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 $\hat{I}\mu >0$, every Parseval frame $\{f_i{\}}_{i=1}^M$ for an $N$-dimensional Hilbert space $H_N$ has a subset of fewer than $(1+\hat{I}\mu )N$ elements which is a frame for $H_N$ with lower frame bound $g(\hat{I}\mu /(2\frac{M}{N}-1))$. 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].

35 pages. Formulation of the Gabor results strengthened. An acknowledgement section added. To appear in the Israel Journal of Mathematics