Completion versus removal of redundancy by perturbation

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=∑∞_k=1ckgk for some scalar coefficients c_k. 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 g_k→0 as k→∞, as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.