Affine and quasi-affine frames for rational dilations

Publication: Research - peer-reviewJournal article – Annual report year: 2011

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In this paper we extend the investigation of quasi-affine systems, which were originally introduced by Ron and Shen [J. Funct. Anal. 148 (1997), 408-447] for integer, expansive dilations, to the class of rational, expansive dilations. We show that an affine system is a frame if, and only if, the corresponding family of quasi-affine systems are frames with uniform frame bounds. We also prove a similar equivalence result between pairs of dual affine frames and dual quasi-affine frames. Finally, we uncover some fundamental differences between the integer and rational settings by exhibiting an example of a quasi-affine frame such that its affine counterpart is not a frame.
Original languageEnglish
JournalAmerican Mathematical Society. Transactions
Publication date2011
Volume363
Journal number4
Pages1887-1924
ISSN0002-9947
DOIs
StatePublished
CitationsWeb of Science® Times Cited: 2

Keywords

  • Rational dilations, Oversampling, Quasi-affine systems, Shift invariant systems, Affine systems, Wavelets
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