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   -> Volume 6, Issue 3

Preprint: Refinable distributions on Lie groups
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Author Message (Wayne Lawton)

PostPosted: Wed Feb 26, 1997 9:25 am    
Subject: Preprint: Refinable distributions on Lie groups
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#7 Preprint: Refinable distributions on Lie groups

Title: Infinite convolution products and refinable distributions on
Lie groups

Wayne Lawton
Institute of System Science
National University of Singapore
Heng Mui Keng Terrace, Kent Ridge
Singapore 119597

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 $cG$ with respect to left invariant Haar measure
are derived. These conditions are used to construct distributions
$phi$ that satisfy
mbox{$Tphi = phi$}
where $T$ is a refinement operator constructed from a measure $mu$
and a dilation automorphism $A$. The existence of $A$ implies $cG$ is
nilpotent and simply connected and the exponential map is an analytic
homeomorphism. Furthermore, there exists a unique minimal compact
mbox{$cK subset cG$}
such that for any open set $cU$ containing $cK,$ and for any
distribution $f$ on $cG$ with compact support, there exists an
integer $n(cU,f)$ such that
mbox{$n geq n(cU,f)$}
mbox{$hbox{supp}(T^{n}f) subsetcU.$}
If $mu$ is supported on an $A$-invariant uniform subgroup $gG,$ then
$T$ is related, by an intertwining operator, to a transition operator
$W$ on $CC(gG).$ Necessary and sufficient conditions for $T^{n}f$ to
converge to $phi in L^{2}$, and for the mbox{$gG$-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
mbox{$Omega := cK cK^{-1} cap gG.$}
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