Completion versus removal of redundancy by perturbation

Abstract

A sequence {g k } ∞ k=1 in a Hilbert space H has the expansion prop-In this paper we analyze the question whether there exist small norm-perturbations of {g k } ∞ 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 {g k } ∞ k=1 such that g k → 0 as k → ∞, as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.