Characterization of the range of the Radon transform on homogeneous trees

This article contains results on the range of the Radon transform $R$ on the set $\mathcal{H}$ of horocycles of a homogeneous tree $T$. Functions of compact support on $\mathcal{H}$ that satisfy two explicit Radon conditions constitute the image under $R$ of functions of finite support on $T$. Replacing functions on $\mathcal{H}$ by distributions, we extend these results to the non-compact case by adding decay criteria.