Infinite convolution products and refinable distributions on Lie groups

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 $\Omega := \mathcal K \mathcal K^{-1} \cap \Gamma .$