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Transactions of the American Mathematical Society

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Affine and quasi-affine frames for rational dilations

Authors: Marcin Bownik and Jakob Lemvig
Journal: Trans. Amer. Math. Soc. 363 (2011), 1887-1924
MSC (2010): Primary 42C40
Published electronically: November 5, 2010
MathSciNet review: 2746669
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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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Additional Information

Marcin Bownik
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403–1222

Jakob Lemvig
Affiliation: Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303S, DK-2800 Kgs. Lyngby, Denmark
Address at time of publication: Institut für Mathematik, Universität Osnabrück, 49069 Osnabrück, Germany

Keywords: Wavelets, affine systems, quasi-affine systems, rational dilations, shift invariant systems, oversampling
Received by editor(s): September 24, 2008
Published electronically: November 5, 2010
Additional Notes: The first author was partially supported by NSF grant DMS-0653881.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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