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 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 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:
https://doi.org/10.1090/S0002-9947-2010-05260-2

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.