Extensions of Bessel sequences to dual pairs of frames

Ole Christensen, Hong Oh Kim, Rae Young Kim

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

Tight frames in Hilbert spaces have been studied intensively for the past years. In this paper we demonstrate that it often is an advantage to use pairs of dual frames rather than tight frames. We show that in any separable Hilbert space, any pairs of Bessel sequences can be extended to a pair of dual frames. If the given Bessel sequences are Gabor systems in L2(R), the extension can be chosen to have Gabor structure as well. We also show that if the generators of the given Gabor Bessel sequences are compactly supported, we can choose the generators of the added Gabor systems to be compactly supported as well. This is a significant improvement compared to the extension of a Bessel sequence to a tight frame, where the added generator only can be compactly supported in some special cases. We also analyze the wavelet case, and find sufficient conditions under which a pair of wavelet systems can be extended to a pair of dual frames. © 2012 Elsevier Inc. All rights reserved.
Original languageEnglish
JournalApplied and Computational Harmonic Analysis
Volume34
Issue number2
Pages (from-to)224-233
ISSN1063-5203
DOIs
Publication statusPublished - 2013

Keywords

  • Bessel sequences
  • Dual frame pairs
  • Gabor frames
  • Wavelet frames

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