Affine and quasi-affine frames for rational dilations

Marcin Bownik, Jakob Lemvig

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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
    JournalTransactions of the American Mathematical Society
    Issue number4
    Pages (from-to)1887-1924
    Publication statusPublished - 2011


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


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