Probability measures on solenoids corresponding to fractal wavelets
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- by Lawrence W. Baggett, Kathy D. Merrill, Judith A. Packer and Arlan B. Ramsay PDF
- Trans. Amer. Math. Soc. 364 (2012), 2723-2748 Request permission
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.References
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Additional Information
- Lawrence W. Baggett
- Affiliation: Department of Mathematics, Campus Box 395, University of Colorado, Boulder, Colorado 80309-0395
- Email: baggett@colorado.edu
- Kathy D. Merrill
- Affiliation: Department of Mathematics, Colorado College, Colorado Springs, Colorado 80903-3294
- Email: kmerrill@coloradocollege.edu
- Judith A. Packer
- Affiliation: Department of Mathematics, Campus Box 395, University of Colorado, Boulder, Colorado 80309-0395
- MR Author ID: 135125
- Email: packer@colorado.edu
- Arlan B. Ramsay
- Affiliation: Department of Mathematics, Campus Box 395, University of Colorado, Boulder, Colorado 80309-0395
- Email: ramsay@colorado.edu
- Received by editor(s): May 20, 2021
- Received by editor(s) in revised form: January 1, 2010
- Published electronically: January 6, 2012
- Additional Notes: This research was supported in part by a grant from the National Science Foundation DMS–0701913
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2888226