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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Infinite convolution products and refinable distributions on Lie groups


Author: Wayne Lawton
Journal: Trans. Amer. Math. Soc. 352 (2000), 2913-2936
MSC (1991): Primary 41A15, 41A58, 42C05, 42C15, 43A05, 43A15
Published electronically: March 2, 2000
MathSciNet review: 1638258
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Abstract:

Sufficient conditions for the convergence in distribution of an infinite convolution product $\mu_1*\mu_2*\ldots$ of measures on a connected Lie group $\mathcal G$ with respect to left invariant Haar measure are derived. These conditions are used to construct distributions $\phi$ that satisfy $T\phi = \phi$where $T$ is a refinement operator constructed from a measure $\mu$and a dilation automorphism $A$. The existence of $A$ implies $\mathcal G$ is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset $\mathcal K \subset \mathcal G$such that for any open set $\mathcal U$ containing $\mathcal K,$ and for any distribution $f$ on $\mathcal G$ with compact support, there exists an integer $n(\mathcal U,f)$ such that $n \geq n(\mathcal U,f)$implies $\hbox{supp}(T^{n}f) \subset\mathcal U.$If $\mu$ is supported on an $A$-invariant uniform subgroup $\Gamma,$ then $T$ is related, by an intertwining operator, to a transition operator $W$ on $\mathbb C(\Gamma).$ Necessary and sufficient conditions for $T^{n}f$ to converge to $\phi \in L^{2}$, and for the $\Gamma$-translates of $\phi$ to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of $W$ to functions supported on $\Omega := \mathcal K \mathcal K^{-1} \cap \Gamma.$


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Additional Information

Wayne Lawton
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: wlawton@math.nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02409-0
PII: S 0002-9947(00)02409-0
Keywords: Lie group, distribution, enveloping algebra, dilation, refinement operator, cascade sequence, transition operator, condition E, Riesz basis
Received by editor(s): March 10, 1997
Received by editor(s) in revised form: April 9, 1998
Published electronically: March 2, 2000
Additional Notes: Research supported in part by the NUS Wavelets Program funded by the National Science and Technology Board and the Ministry of Education, Republic of Singapore.
Article copyright: © Copyright 2000 American Mathematical Society