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On affine frames with transcendental dilations


Author: R. S. Laugesen
Journal: Proc. Amer. Math. Soc. 135 (2007), 211-216
MSC (2000): Primary 42C40; Secondary 42C15
DOI: https://doi.org/10.1090/S0002-9939-06-08456-5
Published electronically: June 29, 2006
MathSciNet review: 2280189
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Abstract: We answer a question of O. Christensen about affine systems in $ L^2(\mathbb{R})$. Specifically, we show that if the dilation factor $ a>1$ is transcendental, then cancellations cannot occur between different scales, in the conditions for the affine system to form a frame. Such cancellations are known to occur when $ a$ is an integer.


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Additional Information

R. S. Laugesen
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: Laugesen@uiuc.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08456-5
Keywords: Affine system, frame, transcendental
Received by editor(s): June 30, 2005
Received by editor(s) in revised form: July 31, 2005
Published electronically: June 29, 2006
Additional Notes: This work was completed during a Visiting Erskine Fellowship at the University of Canterbury, New Zealand, and also with support from National Science Foundation award DMS–0140481.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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