Reconstruction of functions on the sphere from their integrals over hyperplane sections

Abstract

We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point $A$ which lies inside the n-dimensional unit sphere or on the sphere itself. Transforms of the first kind are defined by integration over complete subspheres and can be reduced to the classical Funk transform. Transforms of the second kind perform integration over truncated subspheres, like spherical caps or bowls, and can be reduced to the hyperplane Radon transform. The main tools are analytic families of cosine transforms, Semyanisyi's integrals, and modified stereorgraphic projection with the pole at $A$. Assumptions for functions are close to minimal.