Reducibility of the wavelet representation associated to the Cantor set
HTML articles powered by AMS MathViewer
- by Dorin Ervin Dutkay and Sergei Silvestrov PDF
- Proc. Amer. Math. Soc. 139 (2011), 3657-3664 Request permission
Abstract:
We answer a question by Judith Packer about the irreducibility of the wavelet representation associated to the Cantor set. We prove that if the QMF filter does not have constant absolute value, then the wavelet representation is reducible.References
- Stefan Bildea, Dorin Ervin Dutkay, and Gabriel Picioroaga, MRA super-wavelets, New York J. Math. 11 (2005), 1–19. MR 2154344
- Lawrence W. Baggett, Veronika Furst, Kathy D. Merrill, and Judith A. Packer, Generalized filters, the low-pass condition, and connections to multiresolution analyses, J. Funct. Anal. 257 (2009), no. 9, 2760–2779. MR 2559716, DOI 10.1016/j.jfa.2009.05.004
- Lawrence W. Baggett, Veronika Furst, Kathy D. Merrill, and Judith A. Packer, Classification of generalized multiresolution analyses, J. Funct. Anal. 258 (2010), no. 12, 4210–4228. MR 2609543, DOI 10.1016/j.jfa.2009.12.001
- Lawrence W. Baggett, Nadia S. Larsen, Kathy D. Merrill, Judith A. Packer, and Iain Raeburn, Generalized multiresolution analyses with given multiplicity functions, J. Fourier Anal. Appl. 15 (2009), no. 5, 616–633. MR 2563776, DOI 10.1007/s00041-008-9031-3
- Lawrence W. Baggett, Nadia S. Larsen, Judith A. Packer, Iain Raeburn, and Arlan Ramsay, Direct limits, multiresolution analyses, and wavelets, J. Funct. Anal. 258 (2010), no. 8, 2714–2738. MR 2593341, DOI 10.1016/j.jfa.2009.08.011
- 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, DOI 10.1137/1.9781611970104
- Dorin E. Dutkay and Palle E. T. Jorgensen, Wavelets on fractals, Rev. Mat. Iberoam. 22 (2006), no. 1, 131–180. MR 2268116, DOI 10.4171/RMI/452
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Fourier frequencies in affine iterated function systems, J. Funct. Anal. 247 (2007), no. 1, 110–137. MR 2319756, DOI 10.1016/j.jfa.2007.03.002
- Dorin Ervin Dutkay and Palle E. T. Jorgensen, Martingales, endomorphisms, and covariant systems of operators in Hilbert space, J. Operator Theory 58 (2007), no. 2, 269–310. MR 2358531
- D.E. Dutkay, D.R. Larson, and S. Silvestrov, Irreducible wavelet representations and ergodic automorphisms on solenoids, preprint arXiv:0910.0870v2[math.FA], 2009. To appear in Operators and Matrices.
- Dorin Ervin Dutkay, Harmonic analysis of signed Ruelle transfer operators, J. Math. Anal. Appl. 273 (2002), no. 2, 590–617. MR 1932509, DOI 10.1016/S0022-247X(02)00284-6
- Dorin Ervin Dutkay, Low-pass filters and representations of the Baumslag Solitar group, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5271–5291. MR 2238916, DOI 10.1090/S0002-9947-06-04230-9
- Deguang Han and David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697, x+94. MR 1686653, DOI 10.1090/memo/0697
- Palle E. T. Jorgensen, Ruelle operators: functions which are harmonic with respect to a transfer operator, Mem. Amer. Math. Soc. 152 (2001), no. 720, viii+60. MR 1837681, DOI 10.1090/memo/0720
- Lek-Heng Lim, Judith A. Packer, and Keith F. Taylor, A direct integral decomposition of the wavelet representation, Proc. Amer. Math. Soc. 129 (2001), no. 10, 3057–3067. MR 1840112, DOI 10.1090/S0002-9939-01-05928-7
Additional Information
- Dorin Ervin Dutkay
- Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, P.O. Box 161364, Orlando, Florida 32816-1364
- MR Author ID: 608228
- Email: ddutkay@mail.ucf.edu
- Sergei Silvestrov
- Affiliation: Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden
- Email: ssilvest@maths.lth.se
- Received by editor(s): September 4, 2010
- Published electronically: March 9, 2011
- Additional Notes: Research supported in part by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), the Swedish Research Council 621-2007-6338, the Swedish Royal Academy of Sciences and the Crafoord Foundation.
- Communicated by: Richard Rochberg
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3657-3664
- MSC (2010): Primary 42C40, 28D05, 47A67, 28A80
- DOI: https://doi.org/10.1090/S0002-9939-2011-10913-4
- MathSciNet review: 2813395