Kipriyanov–Radon Transform

A transformation $K_\gamma$ is considered; this transformation is similar to the Radon transform but is adapted to singular differential equations with the Bessel operator $B_{x_n}=\frac {\partial ^2}{\partial x_n^2} +\frac \gamma {x_n}\frac \partial {\partial x_n}$, $\gamma >0$, which is applied with respect to one of the variables. The following formulas are obtained: for the $K_\gamma$ transform of generalized shifts, for the $K_\gamma$ transform of generalized convolutions, a formula for calculating the $K_\gamma$ transform of a homogeneous linear singular differential operator with constant coefficients such that the operator $B_{x_n}$ acts in the last variable, and a formula for the action of this operator on the $K_\gamma$ transform of a test function. The main results of the paper are formulas for reconstructing functions from their $K_\gamma $ transforms. Three cases are considered: (a) the general case of $\gamma>0$, (b) the case when $\gamma>0$ is integer and $n+\gamma$ is odd, and (c) the case when $\gamma>0$ is integer and $n+\gamma $ is even. In case (a), inversion is obtained by applying mixed B-hypersingular integrals. In cases (b) and (c), integer positive powers of the Laplace–Bessel operator $\Delta _{\mathrm B}=\Delta _{x'}+B_{x_n}$ are applied, where $\Delta _{x'}$ is the Laplace operator in the variables $x'=(x_1,\dots ,x_{n-1})$.