Restoring the function set by integrals for the family of parabolas on the plane
Integral geometry is one of the most important sections of the theory of ill-posed problems of mathematical physics and analysis. The urgency of the problems of integral geometry is due to the development of tomographic methods, which raise the requirements for the depth of the applied results, the fact that the solution of problems of integral geometry reduces a number of multidimensional inverse problems for partial differential problems, as well as the internal development needs of the theory of ill-posed problems of mathematical physics and analysis. In this work we consider the problem of reconstructing a 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 posed is weakly ill-posed, that is, stability estimates are obtained in spaces of finite smoothness.