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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Infinite convolution products and refinable distributions on Lie groups
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by Wayne Lawton PDF
Trans. Amer. Math. Soc. 352 (2000), 2913-2936 Request permission

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 $\operatorname {supp}(T^{n}f) \subset \mathcal U.$ 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
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 2913-2936
  • MSC (1991): Primary 41A15, 41A58, 42C05, 42C15, 43A05, 43A15
  • DOI: https://doi.org/10.1090/S0002-9947-00-02409-0
  • MathSciNet review: 1638258