• Corpus ID: 15065768

Redundancy for localized and Gabor frames

@article{Balan2009RedundancyFL,
  title={Redundancy for localized and Gabor frames},
  author={Radu V. Balan and Peter G. Casazza and Zeph Landau},
  journal={arXiv: Functional Analysis},
  year={2009}
}

• R. Balan, P. Casazza, Zeph Landau
• Published 28 April 2009
• Computer Science
• arXiv: Functional Analysis

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].