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Kipriyanov–Radon Transform
L. N. Lyakhov Voronezh State University
Abstract:
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
$x'=(x_1,\dots ,x_{n-1})$.
UDC:
517.9
Received in September 2004
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