This article is cited in 3 papers

Real, complex and functional analysis

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

S. G. Kazantsev

Sobolev Institute of Mathematics, 4, pr. Koptyuga, Novosibirsk, 630090, Russia

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. 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 ${\mathcal S}$.

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

UDC: 514.7, 517.4, 517.98

MSC: 42A45, 44A12, 44A45, 53A45, 53C65

Received July 4, 2018, published December 14, 2018

Language: English

DOI: 10.33048/semi.2018.15.135

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