Infinite convolution products and refinable distributions on Lie groups

Sufficient conditions for the convergence in distribution of an infinite convolution product p1 * P2 * ... of measures on a connected Lie group 5 with respect to left invariant Haar measure are derived. These conditions are used to construct distributions / that satisfy To = / where T is a refinement operator constructed from a measure t and a dilation automorphism A. The existence of A implies 5 is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset IC C 5 such that for any open set U containing IC, and for any distribution f on G with compact support, there exists an integer n(U, f) such that n > n(U, f) implies supp(Tnf) C U. If p is supported on an A-invariant uniform subgroup r, then T is related, by an intertwining operator, to a transition operator W on C((r). Necessary and sufficient conditions for Tnf to converge to / E L2, and for the F-translates of / 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 Q K:= C 'Cn r.

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