Infinite convolution products and refinable distributions on Lie groups期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Infinite convolution products and refinable distributions on Lie groups

Authors:	Wayne Lawton

Institution:	Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543

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 is a refinement operator constructed from a measure $\mu$ and a dilation automorphism . The existence of 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 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 -invariant uniform subgroup $\Gamma,$ then is related, by an intertwining operator, to a transition operator 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 to functions supported on $\Omega := \mathcal K \mathcal K^{-1} \cap \Gamma.$

Keywords:	Lie group distribution enveloping algebra dilation refinement operator cascade sequence transition operator condition E Riesz basis

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