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

摘要

A formula relating the Radon transform of functions of spherical symmetries to the Radon-Kipriyanov transform K_γ fora natural multi-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the K_γ-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.