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. 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 $\mathcal{S}$.