Range of the horocyclic radon transform on trees

In this paper we study the Radon transform R on the set IH. of horocycles o f a homogeneous tree T, and describe its image on various function spaces. We show that the functions of compact support on H that satisfy two explici t Radon conditions constitute the image under R of functions of finite supp ort on T. We extend these results to spaces of functions with suitable deca y on T, whose image under R satisfies corresponding decay conditions and co ntains distributions on H that are not defined pointwise. We also show that R is one-to-one on these spaces. Formulas are expressed in an invariant fa shion in terms of a measure on H preserved by the full automorphism group o f T.