On the Duality Principle by Casazza, Kutyniok, and Lammers

Ole Christensen, Hong Oh Kim, Rae Young Kim

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    The R-dual sequences of a frame {f i } i∈I , introduced by Casazza, Kutyniok and Lammers in (J. Fourier Anal. Appl. 10(4):383–408, 2004), provide a powerful tool in the analysis of duality relations in general frame theory. In this paper we derive conditions for a sequence {ω j } j∈I to be an R-dual of a given frame {f i } i∈I . In particular we show that the R-duals {ω j } j∈I can be characterized in terms of frame properties of an associated sequence {n i } i∈I . We also derive the duality results obtained for tight Gabor frames in (Casazza et al. in J. Fourier Anal. Appl. 10(4):383–408, 2004) as a special case of a general statement for R-duals of frames in Hilbert spaces. Finally we consider a relaxation of the R-dual setup of independent interest. Several examples illustrate the results.
    Original languageEnglish
    JournalJournal of Fourier Analysis and Applications
    Volume17
    Issue number4
    Pages (from-to)640-655
    ISSN1069-5869
    DOIs
    Publication statusPublished - 2011

    Keywords

    • Wexler-Raz theorem
    • Gabor system
    • Frame
    • Riesz basis
    • Duality principle

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