The Funk-Radon transform for hyperplane sections through a common point

TL;DR

This paper studies a generalization of the Funk-Radon transform involving hyperplanes through a common point, providing injectivity and range characterization results by relating it to the classical transform.

Contribution

It introduces a new generalized Radon transform for hyperplanes through a point and establishes its injectivity and range properties via its relation to the classical Funk-Radon transform.

Findings

Proved injectivity of the generalized Radon transform.

Derived a range characterization for the transform.

Connected the generalized transform to the classical Funk-Radon transform.

Abstract

The Funk-Radon transform, also known as the spherical Radon transform, assigns to a function on the sphere its mean values along all great circles. Since its invention by Paul Funk in 1911, the Funk-Radon transform has been generalized to other families of circles as well as to higher dimensions. We are particularly interested in the following generalization: we consider the intersections of the sphere with hyperplanes containing a common point inside the sphere. If this point is the origin, this is the same as the aforementioned Funk--Radon transform. We give an injectivity result and a range characterization of this generalized Radon transform by finding a relation with the classical Funk--Radon transform.