Complete transformation of Radon-Kipriyanov. Some properties

The even Radon-Kipriyanov transform (K_g-transform) is suitable for investigating problems with the Bessel singular differential operator B_{ϒi = $\frac{{\partial }^2}{\partial {\chi }_i^2}+\frac{{\gamma }_i}{{\chi }_i}\frac{\partial }{\partial {\chi }_i}, {\gamma }_{i }$>} 0.  In this paper, we introduce the odd Radon-Kipriyanov transform and complete Radon-Kipriyanov transform to investigation more general equations containing odd B‑derivatives $\frac{\partial }{\partial {\chi }_i}{B}_{\gamma i}^k$,  k = 0, 1, 2, ... (in particular, gradients of functions). Formulas of K_ϒ-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 connection was obtained between the complete Radon-Kipriyanov transform with the Fourier transform and the mixed Fourier-Levitan-Kipriyanov-Katrakhov transform. An analogue of Helgason’s support theorem and an analogue of the Paley-Wiener theorem are presented.