Dmitrii Sergeevich Anikonov

Classical inversion formulas for the integral Radon transform assume that the integrand is smooth. However, this restriction does not fully correspond to application of results in probing theory, which is the main area of application of the Radon transform. It would be more natural to assume that jump discontinuities are admissible for integrands. The paper presents several inversion formulas proved by the authors for piecewise continuous functions. The formulas are compared, and preliminary recommendations on their use for numerical algorithms are given.

This paper is devoted to some questions of inversion for the classical and generalized integral Radon transform. The main question is to determine information about the integrand functions if the values of some integrals are known. A feature of the work of the authors of this message is an analysis of the case when the function is integrated according to hyperplanes in finite-dimensional Euclidean space, and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. At the same time, the number of independent variables describing known integrals are smaller than those of the unknown integrand. We consider discontinuous integrands defined specifically introduced pseudo-convex sets. A Stefan-type problem is posed about finding surfaces discontinuities of the integrand function. The work provides formulas based on the application special integro-differential operators to known data and allowing you to solve the assigned tasks.

We introduce the concept of a pseudoconvex set in an odd-dimensional Euclidean space. The inversion formula is obtained for the Radon transform in the case where the integrand is a piecewise continuous function defined on a pseudoconvex set. The result achieved is a generalization of a previously known property proved for smooth functions.

Some questions concerning the inversion of the classical and generalized integral Radon transforms are discussed. The main issue is to determine information about the integrand if the values of some integrals are known. A feature of this work is that a function is integrated over hyperplanes in a finite-dimensional Euclidean space and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. The independent variables describing the known integrals are fewer than those in the unknown integrand. We consider discontinuous integrands defined on specifically introduced pseudoconvex sets. A Stefan-type problem of finding discontinuity surfaces of the integrand is posed. Formulas for solving the problem under study are derived by applying special integro-differential operators to known data.

We study the problem of the integral geometry, in which the functions are integrated over hyperplanes in the n-dimensional Euclidean space, n = 2 ⁢ m + 1 {n=2m+1} . The integrand is the product of a function of n variables called the density and weight function depending on 2 ⁢ n {2n} variables. Such an integration is called here the weighted Radon transform, which coincides with the classical one if the weight function is equal to one. It is proved the uniqueness for the problem of determination of the surface on which the integrand is discontinuous.

The initial-boundary value problem for the wave equation is considered in the three-dimensional half-space. The oscillation process is initiated by the ini