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

S. G. Kazantsev

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1630-1650

Résumé

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 ${\mathcal S}$.

Détail

Keywords: Funk–Minkowski transform, Funk–-Radon transform, spherical convolution of Hilbert type, Fourier multiplier operator, inverse operator, scalar and vector spherical harmonics, tangential spherical vector field, Helmholtz–Hodge decomposition.
Mots-clés : surface gradient