Fractional Integrals and Tangency Problems in Integral Geometry

TL;DR

This paper introduces new Radon-type transforms involving fractional integrals over tangent geodesic spheres in various geometries, providing inversion formulas and exploring applications in tomography.

Contribution

It presents novel examples of tangent-based Radon transforms in Euclidean, spherical, and hyperbolic spaces with explicit inversion formulas.

Findings

New tangent-based Radon transforms in multiple geometries

Explicit inversion formulas for these transforms

Discussion of applications and challenges in tomography

Abstract

Many known Radon-type transforms of symmetric (radial or zonal) functions are represented by one-dimensional Riemann-Liouville fractional integrals or their modifications. The present article contains new examples of such transforms in the Euclidean, spherical, and hyperbolic settings, when integration is performed over lower-dimensional geodesic spheres or cross-sections, which are tangent to a given surface. Simple inversion formulas are obtained and admissible singularities at the tangency points are studied. Possible applications to the half-ball screening in mathematical tomography and some difficulties related to the general (not necessarily symmetric) case are discussed.