Low-pass filters and representations of the Baumslag Solitar group
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Abstract:
We analyze representations of the Baumslag Solitar group \[ BS(1,N)=\langle u,t | utu^{-1}=t^N\rangle \] that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.References
- Stefan Bildea, Dorin Ervin Dutkay, and Gabriel Picioroaga, MRA super-wavelets, New York J. Math. 11 (2005), 1–19. MR 2154344
- 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, The wavelet Galerkin operator, J. Operator Theory 51 (2004), no. 1, 49–70. MR 2055804
- Dorin Ervin Dutkay, Positive definite maps, representations and frames, Rev. Math. Phys. 16 (2004), no. 4, 451–477. MR 2065233, DOI 10.1142/S0129055X04002047
- D.E. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, to appear in Revista Matematica Iberoamericana.
- Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
- Gernot Greschonig and Klaus Schmidt, Ergodic decomposition of quasi-invariant probability measures. part 2, Colloq. Math. 84/85 (2000), no. part 2, 495–514. Dedicated to the memory of Anzelm Iwanik. MR 1784210, DOI 10.4064/cm-84/85-2-495-514
- Richard F. Gundy, Low-pass filters, martingales, and multiresolution analyses, Appl. Comput. Harmon. Anal. 9 (2000), no. 2, 204–219. MR 1777126, DOI 10.1006/acha.2000.0320
- 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
- P.E.T. Jorgensen, Processes, wavelets and random walks on branches, preprint.
- 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
- Florian Martin and Alain Valette, Markov operators on the solvable Baumslag-Solitar groups, Experiment. Math. 9 (2000), no. 2, 291–300. MR 1780213
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
- E. Weber, Discrete wavelet transforms and $\sigma$-admissible group representations, preliminary version.
Additional Information
- Dorin Ervin Dutkay
- Affiliation: Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
- MR Author ID: 608228
- Email: ddutkay@math.rutgers.edu
- Received by editor(s): July 21, 2004
- Published electronically: July 21, 2006
- Additional Notes: This work was supported in part by NSF grant DMS0457491
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 5271-5291
- MSC (2000): Primary 42C40, 28A78, 46L45, 28D05, 22D25
- DOI: https://doi.org/10.1090/S0002-9947-06-04230-9
- MathSciNet review: 2238916