Israel Journal of Mathematics

, 185:445

Redundancy for localized frames

Article

DOI: 10.1007/s11856-011-0118-1

Cite this article as:
Balan, R., Casazza, P. & Landau, Z. Isr. J. Math. (2011) 185: 445. doi:10.1007/s11856-011-0118-1

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 1-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 ε =s> 0, every Parseval frame {fi}i=1M for an N-dimensional Hilbert space HN has a subset of fewer than (1+ε)N elements which is a frame for HN 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].

Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of MissouriColumbiaUSA
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA