Pizzetti formulae and the Radon Transform on the Sphere

TL;DR

This paper develops Pizzetti-type formulas on the sphere and applies them to invert the spherical Radon transform, providing explicit inversion formulas using invariant differential operators for integrals over sub-spheres and caps.

Contribution

It introduces new Pizzetti formulae on the sphere and uses them to derive explicit inversion formulas for the spherical Radon transform.

Findings

Derived Pizzetti formulae for integrals over sub-spheres and caps.

Expressed integrals as actions of SO(m-1)-invariant differential operators.

Obtained explicit Radon transform inversion formulas on the sphere.

Abstract

In this paper, we obtain Pizzetti-type formulae on regions of the the unit sphere $\mathbb{S}^{m-1}$ of $\mathbb{R}^m$, and study their applications to the problem of inverting the spherical Radon transform. In particular, we approach integration over $(m-2)$-dimensional sub-spheres of $\mathbb{S}^{m-1}$, $(m-1)$-dimensional sub-balls, and over $(m-1)$-dimensional spherical caps as the action of suitable concentrated delta distributions. In turn, this leads to Pizzetti formulae that express such integrals in terms of the action of SO$(m-1)$-invariant differential operators. In the last section of the paper, we use some of these expressions to derive the inversion formulae for the Radon transform on $\mathbb{S}^{m-1}$ in a direct way.