On the Funk-Radon-Helgason Inversion Method in Integral Geometry

TL;DR

This paper explores the Funk-Radon-Helgason inversion method for reconstructing functions from their Radon transforms on constant curvature spaces, introducing new formulas with fractional integrals and emphasizing differentiation operator choices.

Contribution

It provides new inversion formulas involving Erdélyi-Kober fractional integrals and discusses optimal differentiation operators for function reconstruction.

Findings

Derived new inversion formulas with fractional integrals.

Analyzed the impact of differentiation operator choices.

Extended applicability to continuous and L^p functions.

Abstract

The paper deals with totally geodesic Radon transforms on constant curvature spaces. We study applicability of the historically the first Funk-Radon-Helgason method of mean value operators to reconstruction of continuous and $L^p$ functions from their Radon transforms. New inversion formulas involving Erd\'elyi-Kober type fractional integrals are obtained. Particular emphasis is placed on the choice of the differentiation operator in the spirit of the recent Helgason's formula.