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

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. The presented 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 ${\mathcal S}$.