Null Spaces of Radon Transforms

TL;DR

This paper provides new descriptions of the null spaces of various Radon transforms in integral geometry, using advanced integral techniques to extend understanding across multiple geometric contexts.

Contribution

It introduces novel descriptions of null spaces for several Radon transforms, utilizing new results on Gegenbauer-Chebyshev integrals and extending to dual and modified transforms.

Findings

New null space characterizations for hyperplane Radon transform

Descriptions for transforms on spheres and hyperbolic spaces

Extensions to dual and modified transforms

Abstract

We obtain new descriptions of the null spaces of several projectively equivalent transforms in integral geometry. The paper deals with the hyperplane Radon transform, the totally geodesic transforms on the sphere and the hyperbolic space, the spherical slice transform, and the Cormack-Quinto spherical mean transform for spheres through the origin. The consideration extends to the corresponding dual transforms and the relevant exterior/interior modifications. The method relies on new results for the Gegenbauer-Chebyshev integrals, which generalize Abel type fractional integrals on the positive half-line.