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
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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.

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