WEAKLY ILL-POSED PROBLEMS OF INTEGRAL GEOMETRY ON THE PLANE WITH PERTURBATION

Аннотация

Integral geometry is the branch of geometrical analysis that analyzes integral transformations of geometrical nature. More specifically, integral geometry is dealing with properties of functions that can be determined by transformations integrating the function over subsets. In this present paper, we consider a problem of reconstructing of function from a family of parabolas in the upper half-plane with a weight function of a new kind. The uniqueness and existence theorems of the solution of the problem are proved and the inversion formula is derived. It is shown that the solution of the problem is weakly ill-posed, that is, stability estimates are obtained in spaces of finite smoothness. Further, the corresponding problem of integral geometry with perturbation is considered. An uniqueness the- orem for its solution in the class of smooth compactly supported functions with support in a strip and an estimate for the stability of the solution in Sobolev’s spaces are obtained.