Frames generated by actions of countable discrete groups
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- by Kjetil Røysland
- Trans. Amer. Math. Soc. 363 (2011), 95-108
- DOI: https://doi.org/10.1090/S0002-9947-2010-05260-2
- Published electronically: August 11, 2010
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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.References
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Bibliographic 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
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 95-108
- MSC (2010): Primary 42C15, 42C40, 19A13
- DOI: https://doi.org/10.1090/S0002-9947-2010-05260-2
- MathSciNet review: 2719673