Linear problem of integral geometry with smooth weight functions and perturbation

We study two problems of integral geometry in a strip on a family of line segments with a given weight function. In the first case, we consider the problem of reconstruction a function in a strip, if we know the integrals of the sought function on the family of line segments with a given weight function of a special kind. An analytical representation of a solution in the class of smooth finite functions is obtained and the uniqueness and existence theorems for a solution of the problem are proved. A stability estimate of solution in Sobolev spaces is presented, which implies its weakly ill-posedness. For the problem with perturbation the uniqueness theorem and stability estimate of solution were obtained. In the second case, we considered the problem of reconstructing a function given by integral data on the family of line segments with a weight function of exponential type. The uniqueness and existence theorems of a solution are proved. A simple representation of a solution in the class of smooth finite functions is constructed. Next, we consider the corresponding problem of integral geometry with perturbation. The uniqueness theorem in the class of smooth finite functions in a strip is proved and a stability estimate of a solution in Sobolev spaces is received.