On two families of Funk-type transforms

We consider two families of Funk-type transforms that assign to a function on the unit sphere the integrals of that function over spherical sections by planes of fixed dimension. Transforms of the first kind are generated by planes passing through a fixed center outside the sphere. Similar transforms with interior center and with center on the sphere itself we studied in previous publications. Transforms of the second kind, or the parallel slice transforms, correspond to planes that are parallel to a fixed direction. We show that the Funk-type transforms with exterior center express through the parallel slice transforms and the latter are intimately related to the Radon–John d-plane transforms on the Euclidean ball. These results allow us to investigate injectivity of our transforms and obtain inversion formulas for them. We also establish connection between the Funk-type transforms of different dimensions with arbitrary center.