The problem of an unknown boundary for generalized Radon transforms in even-dimensional space

We study the problem of integral geometry, in the case when functions depending on $2n$ variables are integrated over hyperplanes in $n$-dimensional Euclidean space. Such an integration is called here the generalized Radon transform, which coincides with the classical one if the integrand depends on only on $n$ integration variables. In a broad sense, the problem of integral geometry consists in obtaining information about the integrand by values some set of integrals. Here the task is to determination of discontinuity surfaces of the integrand. The uniqueness of the solution is proved, the formula is obtained and the corresponding algorithm is proposed. The results of this work may be used in the theory and practice of probing.

УДК 517.44

Keywords: generalized Radon transform, integral geometry, probing, tomography, differential equation, discontinuous functions.