Probability measures on solenoids corresponding to fractal wavelets

Baggett, Lawrence; Merrill, Kathy; Packer, Judith; Ramsay, Arlan

doi:10.1090/S0002-9947-2012-05584-X

Probability measures on solenoids corresponding to fractal wavelets

Authors: Lawrence W. Baggett, Kathy D. Merrill, Judith A. Packer and Arlan B. Ramsay
Journal: Trans. Amer. Math. Soc. 364 (2012), 2723-2748
MSC (2010): Primary 42C40; Secondary 22D30, 28A80
DOI: https://doi.org/10.1090/S0002-9947-2012-05584-X
Published electronically: January 6, 2012
MathSciNet review: 2888226
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen (2007) is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between solenoid measures associated with classical filters in $\mathbb{R}^d$ and those associated with filters on inflated fractal sets. In particular, it is shown that the classical case produces atomic fiber measures, and as a result supports both suitably defined solenoid MSF wavelets and systems of imprimitivity for the corresponding wavelet representation of the generalized Baumslag-Solitar group. In contrast, the fiber measures for filters on inflated fractal spaces cannot be atomic, and thus can support neither MSF wavelets nor systems of imprimitivity.

1. Lawrence W. Baggett, Jennifer E. Courter, and Kathy D. Merrill, The construction of wavelets from generalized conjugate mirror filters in 𝐿²(ℝⁿ), Appl. Comput. Harmon. Anal. 13 (2002), no. 3, 201–223. MR 1942742, https://doi.org/10.1016/S1063-5203(02)00509-2
2. L. W. Baggett, P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer, Construction of Parseval wavelets from redundant filter systems, J. Math. Phys. 46 (2005), no. 8, 083502, 28. MR 2165848, https://doi.org/10.1063/1.1982768
3. 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, https://doi.org/10.1007/s00041-008-9031-3
4. 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, https://doi.org/10.1016/j.jfa.2009.08.011
5. Marcin Bownik, The construction of 𝑟-regular wavelets for arbitrary dilations, J. Fourier Anal. Appl. 7 (2001), no. 5, 489–506. MR 1845100, https://doi.org/10.1007/BF02511222
6. Ola Bratteli and Palle E. T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (1999), no. 663, x+89. MR 1469149, https://doi.org/10.1090/memo/0663
7. Berndt Brenken, The local product structure of expansive automorphisms of solenoids and their associated 𝐶*-algebras, Canad. J. Math. 48 (1996), no. 4, 692–709. MR 1407604, https://doi.org/10.4153/CJM-1996-036-4
8. Jonas D’Andrea, Kathy D. Merrill, and Judith Packer, Fractal wavelets of Dutkay-Jorgensen type for the Sierpinski gasket space, Frames and operator theory in analysis and signal processing, Contemp. Math., vol. 451, Amer. Math. Soc., Providence, RI, 2008, pp. 69–88. MR 2422242, https://doi.org/10.1090/conm/451/08758
9. 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, https://doi.org/10.1090/S0002-9947-06-04230-9
10. Dorin E. Dutkay and Palle E. T. Jorgensen, Wavelets on fractals, Rev. Mat. Iberoam. 22 (2006), no. 1, 131–180. MR 2268116, https://doi.org/10.4171/RMI/452
11. Dorin Ervin Dutkay and Palle E. T. Jorgensen, Hilbert spaces built on a similarity and on dynamical renormalization, J. Math. Phys. 47 (2006), no. 5, 053504, 20. MR 2239365, https://doi.org/10.1063/1.2196750
12. 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
13. D.E. Dutkay, D.R. Larson, and S. Silvestrov, ArXiv paper, To Appear, 2010.
14. Edward G. Effros, Global structure in von Neumann algebras, Trans. Amer. Math. Soc. 121 (1966), 434–454. MR 192360, https://doi.org/10.1090/S0002-9947-1966-0192360-9
15. R. A. Gopinath and C. S. Burrus, Wavelet transforms and filter banks, Wavelets, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 603–654. MR 1161264
16. John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, https://doi.org/10.1512/iumj.1981.30.30055
17. 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, https://doi.org/10.1090/memo/0720
18. Palle E. T. Jorgensen, Analysis and probability: wavelets, signals, fractals, Graduate Texts in Mathematics, vol. 234, Springer, New York, 2006. MR 2254502
19. Nadia S. Larsen and Iain Raeburn, From filters to wavelets via direct limits, Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 35–40. MR 2270249, https://doi.org/10.1090/conm/414/07797
20. 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, https://doi.org/10.1090/S0002-9939-01-05928-7
21. George W. Mackey, Induced representations of locally compact groups. II. The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953), 193–221. MR 56611, https://doi.org/10.2307/1969786
22. Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases of 𝐿²(𝑅), Trans. Amer. Math. Soc. 315 (1989), no. 1, 69–87. MR 1008470, https://doi.org/10.1090/S0002-9947-1989-1008470-5
23. K. R. Parthasarathy, Introduction to probability and measure, The Macmillan Co. of India, Ltd., Delhi, 1977. MR 0651012
24. Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. MR 833286
25. Robert S. Strichartz, Construction of orthonormal wavelets, Wavelets: mathematics and applications, Stud. Adv. Math., CRC, Boca Raton, FL, 1994, pp. 23–50. MR 1247513
26. Eric Weber, On the translation invariance of wavelet subspaces, J. Fourier Anal. Appl. 6 (2000), no. 5, 551–558. MR 1781094, https://doi.org/10.1007/BF02511546