Pizzetti formulae and the Radon Transform 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.

22 pages