Completion versus removal of redundancy by perturbation
Abstract
A sequence \gk\k=1∞ in a Hilbert space H has the expansion property if each f∈ span \gk\k=1∞ has a representation f= Σk=1∞ ck gk for some scalar coefficients ck. In this paper we analyze the question whether there exist small norm-perturbations of \gk\k=1∞ which allow to represent all f∈ H; the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames \gk\k=1∞ such that gk 0 as k ∞, as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.
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