On R-Duals and the Duality Principle in Gabor Analysis

Diana T. Stoeva, Ole Christensen

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Abstract

The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three authors were actually able to show that the duality principle is a special case of general results for R-duals. In this paper we introduce various alternative R-duals, with focus on what we call R-duals of type II and III. We show how they are related and provide characterizations of the R-duals of type II and III. In particular, we prove that for tight frames these classes coincide with the R-duals by Casazza et al., which is desirable in the sense that the motivating case of tight Gabor frames already is well covered by these R-duals. On the other hand, all the introduced types of R-duals generalize the duality principle for larger classes of Gabor frames than just the tight frames and the Riesz bases; in particular, the R-duals of type III cover the duality principle for all Gabor frames.
Original languageEnglish
JournalJournal of Fourier Analysis and Applications
Volume21
Issue number2
Pages (from-to)383-400
ISSN1069-5869
DOIs
Publication statusPublished - 2015

Keywords

  • MATHEMATICS,
  • Frames
  • Riesz bases
  • R-duals
  • R-duals of type II
  • R-duals of type III
  • Riesz sequences
  • Duality principle
  • HASH(0x4f81490)
  • 42C15

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