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The support theorem for the Radon transform was obtained by L. Helgason. The Radon transform of functions of variables related by spherical symmetry is a special case of a more general transformation, namely, the Radon–Kipriyanov transform K훾. This transformation corresponds to a weight multi-index 훾 = (훾1,... , 훾m) and coincides with the Radon transform if all components of the multi-index 훾 are natural numbers. In general, the K훾-transformation can be interpreted as a transformation of functions of a fraction-dimensional argument. In this paper, we prove a general support theorem. In a special case where 훾 = 0, this theorem coincides with the Helgason theorem. [ABSTRACT FROM AUTHOR]

We consider partial integral in the classes of functions with the mixed Lp– and supnorms or in the anisotropic Lebesgue spaces. For second kind Fredholm integral equations with partial integrals we construct solutions in the form of the operator Neumann series by the method of successive approximations. [ABSTRACT FROM AUTHOR]

For the Kipriyanov operator written in the form of the sum of singular differential Bessel operators with, generally speaking, negative parameters, we obtain a representation in spherical coordinates (the Kipriyanov–Beltrami operator). The operator on the sphere and the corresponding spherical functions ( -harmonics) are introduced. The main properties of the operator on the sphere and the differential equation of -harmonics are provided. A solution of the inner singular Dirichlet problem in a ball centered at the origin in is given. The solution is obtained by the Fourier method in the form of Laplace series in -harmonics. The solution is bounded only in the case where all parameters of the Bessel operators occurring in belong to the interval , , . [ABSTRACT FROM AUTHOR]

We study invertibility of the Radon Kγ-transform (Kipriyanov-Radon transform) with a natural parameter |γ| and prove an analog of the Plancherel formula. [ABSTRACT FROM AUTHOR]

The article examines the Radon-Kipriyanov transform (RKγ-transform) of radial functions, one-variable functions and weighted spherical means of multivariable functions. It explores the use of fractional differentiation operator with an order generated by the weight index to derive the inversion formulas for RKγ-transform of weighted spherical means, and the influence of the dimension of the Euclidean space on such formulas. Moreover, it analyzes the significant relationship between the Abel transform and the RKγ-transform of particular functions.