Recovering functions defined on the unit sphere by integration on a special family of sub-spheres

Abstract

The aim of this article is to derive a reconstruction formula for the recovery of $C^{1}$ functions, defined on the unit sphere ${{\mathbb {S}}}^{n - 1}$, given their integrals on a special family of $n - 2$ dimensional sub-spheres. For a fixed point $\overline{a}$ strictly inside ${{\mathbb {S}}}^{n - 1}$, each sub-sphere in this special family is obtained by intersection of ${{\mathbb {S}}}^{n - 1}$ with a hyperplane passing through $\overline{a}$. The case $\overline{a} = 0$ results in an inversion formula for the special case of integration on great spheres (i.e., Funk transform). The limiting case where $p\in {{\mathbb {S}}}^{n - 1}$ and   $\overline{a}\rightarrow p$ results in an inversion formula for the special case of integration on spheres passing through a common point in ${{\mathbb {S}}}^{n - 1}$.