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

摘要

The Funk–Minkowski transform F associates a function f on the sphere 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, Ff and F.f, are known. Another result of this article is related to the problem of Helmholtz–Hodge decomposition for tangent vector field on the sphere S 2 . We proposed solution for this problem which is used the Funk–Minkowski transform F and Hilbert type spherical convolution S.