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A formula of eigenfunction expansions. I: Case of asymptotic trees. (English) Zbl 0619.60007
In this note we present a new method of giving eigenfunction expansions on a discrete set, namely a connected graph with infinitely many vertices. Our method is to use Poisson kernels which are defined as limits of the quotient of Green kernels having different sources. This has been successfully applied to the case of symmetric spaces and free groups [see e.g.: A. FigĂ -Talamanca and M. A. Picardello, Harmonic analysis on free groups. (1983; Zbl 0536.43001)]. By technical reason we restrict ourselves to the case of asymptotic trees. Details will be published elsewhere.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
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