Recovering Functions Defined on $\Bbb S^{n - 1}$ by Integration on Subspheres Obtained from Hyperplanes Tangent to a Spheroid

TL;DR

This paper introduces a novel method for reconstructing functions on the sphere using integrals over subspheres generated by hyperplanes tangent to a spheroid, with a special case for subspheres passing through a common point.

Contribution

It presents a new approach to spherical function recovery via tangent hyperplanes to a spheroid, including a limiting case for subspheres through a point.

Findings

Method successfully reconstructs functions on the sphere.

Includes a limiting case for subspheres passing through a point.

Provides a framework for inverse spherical transforms.

Abstract

The aim of this article is to introduce a method for recovering functions, defined on the $n - 1$ dimensional unit sphere $\Bbb S^{n - 1}$, using their spherical transform, which integrates functions on $n - 2$ dimensional subspheres, on a prescribed family of subspheres of integration. This family of subspheres is obtained as follows, we take a spheroid $\Sigma$ inside $\Bbb S^{n - 1}$ which contains the points $\pm e_{n}$ and then each subsphere of integration is obtained by the intersection of a hyperplane, which is tangent to $\Sigma$, with $\Bbb S^{n - 1}$. In particular, we obtain as a limiting case, by shrinking the spheroid into its main axis, a method for recovering functions in case where the subspheres of integration pass through a common point in $\Bbb S^{n - 1}$.