A Radon Transform on the Cylinder

TL;DR

This paper introduces a new Radon transform on the cylinder, proves its properties including inversion and null space characterization, and relates it to Chebyshev fractional integrals, expanding integral geometry tools.

Contribution

It defines a parametric Radon transform on the cylinder, establishes its inversion formula, null space, and dual transform, linking it to Chebyshev fractional integrals for the first time.

Findings

The transform is continuous on Sobolev functions.

An explicit inversion formula is derived.

Null space and dual transform are characterized.

Abstract

We define a parametric Radon transform $R$ that assigns to a Sobolev function on the cylinder $\mathbb{S}\times \mathbb{R}$ in $\mathbb{R}^3$ its mean values along sets $E_\zeta$ formed by the intersections of planes through the origin and the cylinder. We show that $R$ is a continuous operator, prove an inversion formula, provide a support theorem, as well as a characterization of its null space. We conclude by presenting a formula for the dual transform $R^*$. We show that $R$ and its dual $R^*$ are related to the right-sided and left-sided Chebyshev fractional integrals. Using this relationship, we characterize the null space of $R$ and $R^*$ and provide an inversion formula for $R^*$.