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

✍ Scribed by Alexander Katsevich

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 195 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310251

✦ Synopsis

Abstract

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)