An integral geometry underdetermined problem for a family of curves

Abstract

In a general integral geometry problem, there are given the integrals of an unknown function over certain manifolds. The traditional statement of the problem consists in determining the integrand. We consider the case of an underdetermined problem when the unknown functions depend on a greater number of variables than the given integrals. These situations appear in a few applied problems when a rather complicated mathematical model is used and no a priori information is available. For overcoming the lack of appropriate data, we pose the problem of finding part of the information unknown; namely, we search only for the discontinuity surfaces of the integrand. The corresponding uniqueness theorem is proved. The present paper finalizes our studies into the case of integration over one-dimensional manifolds. In the previous articles we considered similar problems in the case of integration over straight lines. In this paper the same result is proved for the integration of unknown functions over unknown curves.