Sparsity and spectral properties of dual frames

Felix Krahmer, Gitta Kutyniok, Jakob Lemvig

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    Abstract

    We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than $n^2$ non-vanishing entries, where $n$ denotes the ambient dimension, and that for most frames no sparser dual is possible. Moreover, we derive an expression for the exact sparsity level of the sparsest dual for any given finite frame using a generalized notion of spark. We then study the spectral properties of dual frames in terms of singular values of the synthesis operator. We provide a complete characterization for which spectral patterns of dual frames are possible for a fixed frame. For many cases, we provide simple explicit constructions for dual frames with a given spectrum, in particular, if the constraint on the dual is that it be tight.
    Original languageEnglish
    JournalLinear Algebra and Its Applications
    Volume439
    Issue number4
    Pages (from-to)982–998
    ISSN0024-3795
    DOIs
    Publication statusPublished - 2013
    Event17th Conference of the International Linear Algebra Society (ILAS 2011) - Braunschweig, Germany
    Duration: 22 Aug 201126 Aug 2011
    http://134.169.92.18/ilas/

    Conference

    Conference17th Conference of the International Linear Algebra Society (ILAS 2011)
    CountryGermany
    CityBraunschweig
    Period22/08/201126/08/2011
    Internet address

    Keywords

    • Dual frames
    • Frame theory
    • Singular values
    • Sparse duals
    • Sparsity
    • Spectrum
    • Tight frames

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