Frames generated by actions of countable discrete groups
Author:
Kjetil Røysland
Journal:
Trans. Amer. Math. Soc. 363 (2011), 95108
MSC (2010):
Primary 42C15, 42C40, 19A13
Published electronically:
August 11, 2010
MathSciNet review:
2719673
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References 
Similar Articles 
Additional Information
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 shiftinvariant spaces and wavelet theory. One of the main findings in this paper is that whenever a shiftinvariant subspace in has compactly supported dual frame generators, then it also has compactly supported biorthogonal 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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 Ola Bratteli and Derek W. Robinson.
Operator algebras and quantum statistical mechanics. 1. Texts and Monographs in Physics. SpringerVerlag, New York, second edition, 1987.  and algebras, symmetry groups, decomposition of states. MR 887100 (88d:46105)
 [CDF92]
 A. Cohen, Ingrid Daubechies, and J.C. Feauveau.
Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math., 45(5):485560, 1992. MR 1162365 (93e:42044)
 [Dau92]
 Ingrid Daubechies.
Ten lectures on wavelets, volume 61 of CBMSNSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107 (93e:42045)
 [Dav96]
 Kenneth R. Davidson.
algebras by example, volume 6 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 1996. MR 1402012 (97i:46095)
 [DR07]
 Dorin Ervin Dutkay and Kjetil Røysland.
The algebra of harmonic functions for a matrixvalued transfer operator. J. Funct. Anal., 252(2):734762, 2007. MR 2360935 (2008m:42056)
 [FL02]
 Michael Frank and David R. Larson.
Frames in Hilbert modules and algebras. J. Operator Theory, 48(2):273314, 2002. MR 1938798 (2003i:42040)
 [Gub88]
 I. Dzh. Gubeladze.
The Anderson conjecture and a maximal class of monoids over which projective modules are free. Mat. Sb. (N.S.), 135(177)(2):169185, 271, 1988. MR 937805 (89d:13010)
 [HL00]
 Deguang Han and David R. Larson.
Frames, bases and group representations. Mem. Amer. Math. Soc., 147(697):x+94, 2000. MR 1686653 (2001a:47013)
 [Lam78]
 T. Y. Lam.
Serre's conjecture. Lecture Notes in Mathematics, Vol. 635. SpringerVerlag, Berlin, 1978. MR 0485842 (58:5644)
 [Lam06]
 T. Y. Lam.
Serre's problem on projective modules. Springer Monographs in Mathematics. SpringerVerlag, Berlin, 2006. MR 2235330 (2007b:13014)
 [Lan95]
 E. C. Lance.
Hilbert modules, volume 210 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694 (96k:46100)
 [Lan02]
 Serge Lang.
Algebra, volume 211 of Graduate Texts in Mathematics. SpringerVerlag, New York, third edition, 2002. MR 1878556 (2003e:00003)
 [LR07]
 Nadia S. Larsen and Iain Raeburn.
Projective multiresolution analyses arising from direct limits of Hilbert modules. Math. Scand., 100(2):317360, 2007. MR 2339372 (2008m:42059)
 [Pac07]
 Judith A. Packer.
Projective multiresolution analyses for dilations in higher dimensions. J. Operator Theory, 57(1):147172, 2007. MR 2304920 (2008b:46098)
 [Ped89]
 Gert K. Pedersen.
Analysis now, volume 118 of Graduate Texts in Mathematics. SpringerVerlag, New York, 1989. MR 971256 (90f:46001)
 [PR03]
 Judith A. Packer and Marc A. Rieffel.
Wavelet filter functions, the matrix completion problem, and projective modules over . J. Fourier Anal. Appl., 9(2):101116, 2003. MR 1964302 (2003m:42063)
 [PR04]
 Judith A. Packer and Marc A. Rieffel.
Projective multiresolution analyses for . J. Fourier Anal. Appl., 10(5):439464, 2004. MR 2093911 (2005f:46133)
 [Røy08]
 Kjetil Røysland.
Symmetries in projective multiresolution analyses. J. Fourier Anal. Appl., 14(2):267285, 2008. MR 2383725 (2009i:42016)
 [Swa78]
 Richard G. Swan.
Projective modules over Laurent polynomial rings. Trans. Amer. Math. Soc., 237:111120, 1978. MR 0469906 (57:9686)
 [Woo04]
 Peter John Wood.
Wavelets and Hilbert modules. J. Fourier Anal. Appl., 10(6):573598, 2004. MR 2105534 (2005h:42079)
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Additional Information
Kjetil Røysland
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053, Blindern, NO0316 Oslo, Norway
Address at time of publication:
Department of Biostatistics, University of Oslo, Sognsvannsv. 9, PO Box 1122, Blindern, NO0317 Oslo, Norway
Email:
roysland@math.uio.no
DOI:
http://dx.doi.org/10.1090/S000299472010052602
Keywords:
Frames,
shiftinvariant 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.
