Please use this identifier to cite or link to this item: https://scholarbank.nus.edu.sg/handle/10635/103420
Title: Infinite convolution products and refinable distributions on lie groups
Authors: Lawton, W. 
Keywords: Cascade sequence
Condition e
Dilation
Distribution
Enveloping algebra
Lie group
Refinement operator
Riesz basis
Transition operator
Issue Date: 2000
Citation: Lawton, W. (2000). Infinite convolution products and refinable distributions on lie groups. Transactions of the American Mathematical Society 352 (6) : 2913-2936. ScholarBank@NUS Repository.
Abstract: 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 n 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 Q with compact support, there exists an integer n(U, f) such that n ≥ n(U, f) implies supp(Tnf) C 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 Tn f 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. ©2000 American Mathematical Society.
Source Title: Transactions of the American Mathematical Society
URI: http://scholarbank.nus.edu.sg/handle/10635/103420
ISSN: 00029947
Appears in Collections:Staff Publications

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