The Radon–Kipriyanov Transform of the Generalized Spherical Mean of a Function

A formula relating the Radon transform of functions of spherical symmetries to the Radon–Kipriyanov transform $K_\gamma$ for a natural multi-index $\gamma$ is given. For an arbitrary multi-index $\gamma$, formulas for the representation of the $K_\gamma$-transform of a radial function as fractional integrals of Erdelyi–Kober integral type and of Riemann–Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon–Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables.