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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Frames generated by actions of countable discrete groups


Author: Kjetil Røysland
Journal: Trans. Amer. Math. Soc. 363 (2011), 95-108
MSC (2010): Primary 42C15, 42C40, 19A13
Published electronically: August 11, 2010
MathSciNet review: 2719673
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Abstract: We consider dual frames generated by actions of countable discrete groups on a Hilbert space. Module frames in a class of modules over a group algebra are shown to coincide with a class of ordinary frames in a representation of the group. This has applications to shift-invariant spaces and wavelet theory. One of the main findings in this paper is that whenever a shift-invariant subspace in $ L^2(\mathbb{R}^n)$ has compactly supported dual frame generators, then it also has compactly supported bi-orthogonal generators. The crucial part in the proof is a theorem by Swan that states that every finitely generated projective module over the Laurent polynomials in $ n$ variables is free.


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Additional Information

Kjetil Røysland
Affiliation: Department of Mathematics, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norway
Address at time of publication: Department of Biostatistics, University of Oslo, Sognsvannsv. 9, PO Box 1122, Blindern, NO-0317 Oslo, Norway
Email: roysland@math.uio.no

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05260-2
PII: S 0002-9947(2010)05260-2
Keywords: Frames, shift-invariant subspaces, multiresolution analysis and unitary group representations.
Received by editor(s): June 26, 2008
Published electronically: August 11, 2010
Additional Notes: This research was supported in part by the Research Council of Norway, project number NFR 154077/420. Some of the final work was also done with support from the project NFR 170620/V30.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.