Infinite Convolution Products and Refinable Distributions on Lie Groups

Sufficient conditions for the convergence in distribution of an infinite convolution product μ1 ∗μ2 ∗ . . . of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions φ that satisfy Tφ = φ where T is a refinement operator constructed from a measure μ 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(Tnf) ⊂ U . If μ is supported on an A-invariant uniform subgroup Γ, then T is related, by an intertwining operator, to a transition operator W on C(Γ). Necessary and sufficient conditions for Tnf to converge to φ ∈ L2, and for the Γ-translates of φ 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 Ω := KK−1 ∩ Γ.