On single and paired shifted Funk transform

The classical result due to Funk is about the reconstruction of even functions on 
the unit sphere in $R^n$ from their integrals over the cross-sections by the 
hyperplanes (or k-planes) through the origin. In modern applications in tomography 
and imaging, this transform is involved in reconstruction methods in diffusion MRI. 
Last years, the shifted Funk transform, with the common point (center) of the cross-
sections different form the origin, has been studied by several authors. My talk 
will be devoted to new results in this area. I will touch upon the description of 
the kernel of the shifted Funk transform and its relation with the classical one, 
delivered by action of the pseudo-orthogonal group on the unit real ball. In most 
cases, the kernel is nontrivial, so that inverting the transform is impossible. 
However, it appears that to completely recover a function on the unit sphere a pair 
Funk data may be enough and this possibility depends on the mutual location of the 
centers. The size of the common kernel of a paired transform appears to be related 
with the type of iteration dynamics of certain billiard-like self-mapping of the 
unit sphere and understanding this dynamics yields necessary and sufficient conditions 
of the injectivity of the paired Funk transform. In the cases of injectivity, we 
present a reconstruction procedure in terms of an $L^p$-convergent Neumann type series 
with p in a certain range. It is a joint work with Boris Rubin from Louisiana State University.