Radon transforms on homogeneous trees

Abstract

We study the different horospherical Radon transforms that arise by regarding a homogeneous tree T as a simplicial complex whose simplices are vertices V, edges E or flags F (flags are oriented edges). The ends (infinite geodesic rays starting at a reference vertex) provide a boundary $Ω$ for the tree. Then the horospheres form a trivial principal fiber bundle with base $Ω$ and fiber $\mathZ$. There are three such fiber bundles, consisting of horospheres of vertices, edges or flags, but they are isomorphic: however, no isomorphism between these fiber bundles maps special sections to special sections (a special section consists of the set of horospheres through a given vertex, edge or flag). The groups of automorphisms of the fiber bundles contain a subgroup $A$ of parallel shifts, analogous to the Cartan subgroup of a semisimple group. The normalized eigenfunctions of the Laplace operator on T are boundary integrals of complex powers of the Poisson kernel, that is characters of $A$, and are matrix coefficients of representations induced from $A$ in the sense of Mackey, the so-called spherical representations. The vertex-horospherical Radon transform consists of summation over V in each vertex-horosphere, and similarly for edges or flags. We prove inversion formulas for all these Radon transforms, and give applications to harmonic analysis and the Plancherel measure on T. We show via integral geometry that the spherical representations for vertices and edges are equivalent. Also, we define the Radon back-projections and find the inversion operator of each Radon transform by composing it with its back-projection. This gives rise to a convolution operator on T, whose symbol is obtained via the spherical Fourier transform, and its reciprocal is the symbol of the Radon inversion formula.