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

Abstract

The aim of this article is to introduce a method for recovering functions, defined on the $n-1$ dimensional unit sphere ${\mathbb{S}}^{n-1}$, 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 ${\mathbb{S}}^{n-1}$ 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 ${\mathbb{S}}^{n-1}$. 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 ${\mathbb{S}}^{n-1}$.

arXiv comment14 pages

Cited by in corpus (1)

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