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 } M i=1 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].
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The first author was supported by NSF DMS 0807896.
The second author was supported by NSF DMS 0704216 and 1008183, and thanks the American Institute of Mathematics for their continued support.
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Balan, R., Casazza, P. & Landau, Z. Redundancy for localized frames. Isr. J. Math. 185, 445–476 (2011). https://doi.org/10.1007/s11856-011-0118-1
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DOI: https://doi.org/10.1007/s11856-011-0118-1