On Dual Gabor Frame Pairs Generated by Polynomials

Ole Christensen, Kim Rae Young

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for frames generated by any compactly supported function phi whose integer-translates form a partition of unity, e.g., a B-spline, we construct a class of dual frame generators, formed by linear combinations of translates of phi. This allows us to chose a dual generator with special properties, for example, the one with shortest support, or a symmetric one in case the frame itself is generated by a symmetric function. One of these dual generators has the property of being constant on the support of the frame generator.
    Original languageEnglish
    JournalJournal of Fourier Analysis and Applications
    Volume16
    Issue number1
    Pages (from-to)1-16
    ISSN1069-5869
    DOIs
    Publication statusPublished - 2010

    Keywords

    • Dual frame
    • Strang-Fix conditions
    • B-spline
    • Dual generator
    • Gabor frames
    • Polynomial

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