  The paper deals with totally geodesic Radon transforms on constant curvature
spaces. We study applicability of the historically the first
Funk-Radon-Helgason method of mean value operators to reconstruction of
continuous and $L^p$ functions from their Radon transforms. New inversion
formulas involving Erd\'elyi-Kober type fractional integrals are obtained.
Particular emphasis is placed on the choice of the differentiation operator in
the spirit of the recent Helgason's formula.
