Refinable Distributions on Lie Groups

Suucient conditions for the convergence in distribution of an inn-nite convolution product 1 2 : : : of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions that satisfy T = where T is a reenement operator constructed from a measure and a dilation automorphism A. The existence of A implies G is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset K G such that for any open set U containing K; 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(T n f) U: If is supported on an A-invariant uniform subgroup ?; then T is related, by an intertwining operator , to a transition operator W on C(?): Necessary and suucient conditions for T n f to converge to 2 L 2 , and for the ?-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 := KK ?1 \ ?: 1. Introduction and Statement of Results This paper extends concepts, associated with the theory of reenable functions