Moebius deconvolution on the hyperbolic plane with application to impedance density estimation

In this paper we consider a novel statistical inverse problem on the Poincare, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of 2 x 2 real matrices of determinant one via Mobius 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 Mains transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincare plane exactly describes the physical system that is of statistical interest.