Sparsity and spectral properties of dual frames

Publication: Research - peer-reviewJournal article – Annual report year: 2012

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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
StatePublished - 2013
Peer-reviewedYes

Conference

Conference17th Conference of the International Linear Algebra Society (ILAS 2011)
CountryGermany
CityBraunschweig
Period22/08/201126/08/2011
Internet addresshttp://134.169.92.18/ilas/
CitationsWeb of Science® Times Cited: 2

Keywords

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