Completion versus removal of redundancy by perturbation

Ole Christensen, Marzieh Hasannasab

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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=∑∞k=1ckgk 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.
Original languageEnglish
JournalCanadian Mathematical Bulletin
Number of pages14
ISSN0008-4395
DOIs
Publication statusPublished - 2021

Keywords

  • Frames
  • Riesz bases
  • Completeness
  • Redundancy

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