Affine and quasi-affine frames for rational dilations

Bownik, Marcin; Lemvig, Jakob

doi:10.1090/S0002-9947-2010-05200-6

Affine and quasi-affine frames for rational dilations
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by Marcin Bownik and Jakob Lemvig PDF

Trans. Amer. Math. Soc. 363 (2011), 1887-1924 Request permission

Abstract:

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.

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