Funk-Minkowski transform and spherical convolution of Hilbert type in reconstructing functions on the sphere

TL;DR

This paper introduces an analytical inversion formula for the Funk--Minkowski transform on the sphere, enabling complete reconstruction of functions from their mean values along great circles, and explores related Helmholtz--Hodge decomposition using spherical convolution.

Contribution

It provides a new inversion formula for the Funk--Minkowski transform and applies it to Helmholtz--Hodge decomposition using spherical convolution, advancing spherical function analysis.

Findings

Complete reconstruction of functions from two Funk--Minkowski transforms.

Solution to Helmholtz--Hodge decomposition using Funk--Minkowski and Hilbert spherical convolution.

Analytical inversion formula for Funk--Minkowski transform on the sphere.

Abstract

The Funk--Minkowski transform ${\mathcal F}$ associates a function $f$ on the sphere ${\mathbb S}^2$ with its mean values (integrals) along all great circles of the sphere. Thepresented analytical inversion formula reconstruct the unknown function $f$ completely if two Funk--Minkowski transforms, ${\mathcal F}f$ and ${\mathcal F} \nabla f$, are known. Another result of this article is related to the problem of Helmholtz--Hodge decomposition for tangent vector field on the sphere ${\mathbb S}^2$. We proposed solution for this problem which is used the Funk-Minkowski transform ${\mathcal F}$ and Hilbert type spherical convolution ${\mathcal S}$.