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

Full-text PDF

Abstract | 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 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.

**[BR87]**Ola Bratteli and Derek W. Robinson,*Operator algebras and quantum statistical mechanics. 1*, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1987. 𝐶*- and 𝑊*-algebras, symmetry groups, decomposition of states. MR**887100****[CDF92]**A. Cohen, Ingrid Daubechies, and J.-C. Feauveau,*Biorthogonal bases of compactly supported wavelets*, Comm. Pure Appl. Math.**45**(1992), no. 5, 485–560. MR**1162365**, 10.1002/cpa.3160450502**[Dau92]**Ingrid Daubechies,*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107****[Dav96]**Kenneth R. Davidson,*𝐶*-algebras by example*, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR**1402012****[DR07]**Dorin Ervin Dutkay and Kjetil Røysland,*The algebra of harmonic functions for a matrix-valued transfer operator*, J. Funct. Anal.**252**(2007), no. 2, 734–762. MR**2360935**, 10.1016/j.jfa.2007.04.014**[FL02]**Michael Frank and David R. Larson,*Frames in Hilbert 𝐶*-modules and 𝐶*-algebras*, J. Operator Theory**48**(2002), no. 2, 273–314. MR**1938798****[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)**(1988), no. 2, 169–185, 271 (Russian); English transl., Math. USSR-Sb.**63**(1989), no. 1, 165–180. MR**937805****[HL00]**Deguang Han and David R. Larson,*Frames, bases and group representations*, Mem. Amer. Math. Soc.**147**(2000), no. 697, x+94. MR**1686653**, 10.1090/memo/0697**[Lam78]**T. Y. Lam,*Serre’s conjecture*, Lecture Notes in Mathematics, Vol. 635, Springer-Verlag, Berlin-New York, 1978. MR**0485842****[Lam06]**T. Y. Lam,*Serre’s problem on projective modules*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. MR**2235330****[Lan95]**E. C. Lance,*Hilbert 𝐶*-modules*, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR**1325694****[Lan02]**Serge Lang,*Algebra*, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR**1878556****[LR07]**Nadia S. Larsen and Iain Raeburn,*Projective multi-resolution analyses arising from direct limits of Hilbert modules*, Math. Scand.**100**(2007), no. 2, 317–360. MR**2339372****[Pac07]**Judith A. Packer,*Projective multiresolution analyses for dilations in higher dimensions*, J. Operator Theory**57**(2007), no. 1, 147–172. MR**2304920****[Ped89]**Gert K. Pedersen,*Analysis now*, Graduate Texts in Mathematics, vol. 118, Springer-Verlag, New York, 1989. MR**971256****[PR03]**Judith A. Packer and Marc A. Rieffel,*Wavelet filter functions, the matrix completion problem, and projective modules over 𝐶(𝕋ⁿ)*, J. Fourier Anal. Appl.**9**(2003), no. 2, 101–116. MR**1964302**, 10.1007/s00041-003-0010-4**[PR04]**Judith A. Packer and Marc A. Rieffel,*Projective multi-resolution analyses for 𝐿²(ℝ²)*, J. Fourier Anal. Appl.**10**(2004), no. 5, 439–464. MR**2093911**, 10.1007/s00041-004-3065-y**[Røy08]**Kjetil Røysland,*Symmetries in projective multiresolution analyses*, J. Fourier Anal. Appl.**14**(2008), no. 2, 267–285. MR**2383725**, 10.1007/s00041-008-9010-8**[Swa78]**Richard G. Swan,*Projective modules over Laurent polynomial rings*, Trans. Amer. Math. Soc.**237**(1978), 111–120. MR**0469906**, 10.1090/S0002-9947-1978-0469906-4**[Woo04]**Peter John Wood,*Wavelets and Hilbert modules*, J. Fourier Anal. Appl.**10**(2004), no. 6, 573–598. MR**2105534**, 10.1007/s00041-004-0828-4

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
42C15,
42C40,
19A13

Retrieve articles in all journals with MSC (2010): 42C15, 42C40, 19A13

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

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.