The Funk-Radon transform for hyperplane sections through a common point
For mathematicians studying integral geometry, this is an incremental extension of known theory to a new family of curves.
The paper generalizes the Funk-Radon transform to hyperplane sections through a common point inside the sphere, providing an injectivity result and range characterization by relating it to the classical transform.
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.