The Generalized Mader's Inversion Formulas for the Radon Transforms

TL;DR

This paper generalizes Mader's inversion formulas for Radon transforms to spaces of constant curvature, extending classical results to more general geometric settings using integral geometry and complex analysis.

Contribution

It introduces generalized inversion formulas for the hyperplane and k-plane Radon transforms on constant curvature spaces, expanding their applicability.

Findings

Generalized Mader's formulas to constant curvature spaces

Extended Helgason's inversion formula to arbitrary constant curvature spaces

Combined integral geometry and complex analysis tools for derivations

Abstract

In 1927 Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on $\bbr^n$. These formulas differ from the original ones by Radon and seem to be forgotten. We generalize Mader's formulas to totally geodesic Radon transforms in any dimension on arbitrary constant curvature space. Another new interesting inversion formula for the $k$-plane transform was presented in the recent book "Integral geometry and Radon transform" by S. Helgason. We extend this formula to arbitrary constant curvature space. The paper combines tools of integral geometry and complex analysis.