In this paper we consider a novel statistical inverse problem on the Poincaré, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of 2 × 2 real matrices of determinant one via Möbius transformations. Our approach is based on a deconvolution technique which relies on the Helgason–Fourier calculus adapted to this hyperbolic space. This gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random Möbius transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincaré plane exactly describes the physical system that is of statistical interest.
"Möbius deconvolution on the hyperbolic plane with application to impedance density estimation." Ann. Statist. 38 (4) 2465 - 2498, August 2010. https://doi.org/10.1214/09-AOS783