ABSTRACT
This article contains recent results on injectivity and inversion of the Radon transform R on the set ℋ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187834/69daaf7f-2564-4f2f-9a9d-3a9774ce28b0/content/eq739.tif"/> of horocycles of a homogeneous tree T, as well as on the characterization of its range for different domains. The Radon transform maps the set of functions of finite support on T onto the set of functions of compact support on ℋ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187834/69daaf7f-2564-4f2f-9a9d-3a9774ce28b0/content/eq740.tif"/> that satisfy the Radon conditions. This result holds also in the non-compact case, provided that the appropriate decay criteria are added, but functions on ℋ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187834/69daaf7f-2564-4f2f-9a9d-3a9774ce28b0/content/eq741.tif"/> need to be replaced by distributions. We prove here an inversion formula valid on such space of distributions, generalizing a formula that was previously known (for T not necessarily homogeneous) on the image of L 1(T).
