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 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.
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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
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
DOI:
https://doi.org/10.1090/S0002-9947-2012-05584-X
Keywords:
Fractals,
wavelets,
solenoids,
probability measures
Received by editor(s):
January 11, 2020
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
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.