An inversion formula for the dual horocyclic Radon transform on the hyperbolic plane

Consider the Poincare unit disk model for the hyperbolic plane H 2 . Let Ξ be the set of all horocycles in H 2 parametrized by ( θ, p ), where e iθ is the point where a horocycle ξ is tangent to the boundary | z | = 1, and p is the hyperbolic distance from ξ to the origin. In this paper we invert the dual Radon transform R * : μ ( θ, p ) → $ \check \mu $ ( z ) under the assumption of exponential decay of μ and some of its derivatives. The additional assumption is that P m ( d/dp )( μ m ( p ) e p ) be even for all m ∈ ℤ. Here P m ( d/dp ) is a family of differential operators introduced by Helgason, and μ m ( p ) are the coefficients of the Fourier series expansion of μ ( θ, p ). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)