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

2018/06/05 by Kazantsev, Sergey G.
#44A12 #53A45 #53C65 #65R10 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1806.06672

Abstract

The Funk--Minkowski transform \mathcal F associates a function f on the sphere \mathbb S² 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 Ff and \mathcal 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 \mathbb S². We proposed solution for this problem which is used the Funk-Minkowski transform \mathcal F and Hilbert type spherical convolution \mathcal S.

Related

• Reconstruction of a function defined on a sphere using the Funk transform
• Reconstruction of functions on the sphere from their integrals over hyperplane sections
• The Fourier Transform Approach to Inversion of lambda-Cosine and Funk Transforms on the Unit Sphere
• Exact inversion of Funk-Radon transforms with non-algebraic geometries
• On Two Families of Funk-Type Transforms