We consider the Radon transform for a dual pair $(X,\Xi)$, where $X=G/K$ is noncompact symmetric space and $\Xi$ of horocycles $X$. address unitarization problem that was considered (and solved in some cases) by Helgason, namely determination pseudo-differential operator such pre-composition with extends to unitary $\mathcal{Q}\colon L^2(X)\to L_\flat^2(\Xi)$, $L_\flat^2(\Xi)$ closed subspace $...

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 t...

For a one-parameter family of simple metrics constant curvature ($4\\kappa$ for $\\kappa\\in (-1,1)$) on the unit disk $M$, we first make explicit Pestov–Uhlmann range characterization geodesic X-ray transform, by constructing basis functions making up its and co-kernel. Such also translates into moment conditions $à$ $la$ Helgason–Ludwig or Gel'fand–Graev. We then derive an Singular Value Deco...