@article {Cohen:1998-02-01T00:00:00:0196-8858:253,
author = "Cohen, J.M. and Colonna, F.",
title = "Spectral Analysis on Homogeneous Trees",
journal = "Advances in Applied Mathematics",
volume = "20",
number = "2",
year = "1998-02-01T00:00:00",
abstract = "For each complex number z, we construct an operator H z defined on the space of all complex-valued functions on a homogeneous tree. This operator has the property that if a function f is harmonic (i.e., the local averaging operator fixes the values of f), then H z(f) is z-harmonic (i.e., the local averaging operator multiplies H z(f) by z). Because the Laplacian is the local averaging operator minus the identity, a z-harmonic function is an eigenfunction of the Laplacian relative to the eigenvalue z - 1. We show that all z-harmonic functions are in the image of H z, and we compare H z to another well-known operator which converts harmonic functions to z-harmonic functions. We then study the problem of representing a function as the integral of z-harmonic functions with respect to a distribution. In particular, if a function grows no faster than exponentially, then the distribution is of the form f(z, - ) dlambda(z), where f(z, - ) is a z-harmonic function and lambda is the Lebesgue measure in ℂ. If the base of the growth is sufficiently small, the distribution is supported over the interval . Copyright 1998 Academic Press.",
pages = "253-274",
url = "http://www.ingentaconnect.com/content/ap/am/1998/00000020/00000002/art00570"
}