Complete Radon–Kipriyanov Transform: Some Properties

Abstract

The even Radon–Kipriyanov transform (K_γ-transform) is suitable for studying problems with the Bessel singular differential operator ${{B}_{{{{\gamma }_{i}}}}} = \frac{{{{\partial }^{2}}}}{{\partial x_{i}^{2}}} + \frac{{{{\gamma }_{i}}}}{{{{x}_{i}}~}}\frac{\partial }{{\partial {{x}_{i}}}},{{\gamma }_{i}} > 0$. In this work, the odd Radon–Kipriyanov transform and the complete Radon–Kipriyanov transform are introduced to study more general equations containing odd B-derivatives $\frac{\partial }{{\partial {{x}_{i}}}}~B_{{{{\gamma }_{i}}}}^{k},~~k = 0, 1, 2,~ \ldots$ (in particular, gradients of functions). Formulas of the K_γ-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B.M. Levitan and the odd Bessel transform introduced by I.A. Kipriyanov and V.V. Katrakhov, a relationship of the complete Radon–Kipriyanov transform with the Fourier transform and the mixed Fourier–Levitan–Kipriyanov–Katrakhov transform is deduced. An analogue of Helgason’s support theorem and an analog of the Paley–Wiener theorem are given.