ABSTRACT

This chapter considers the problem of finding a function from its integrals over a family of one-sheeted cones in the n-dimensional Euclidean space of an arbitrary odd dimension. This problem is a Volterra-type integral geometry problem. The chapter provides a stability estimate for a solution to the problem in spaces of finite smoothness, thereby demonstrating weak ill-posedness of the problem, and also construct a representation for a solution. It describes uniqueness theorem and stability estimates for the solution to problem with perturbation under rather general assumptions on the weight function of perturbation. The uniqueness theorem was formulated, a simple representation for the solution was constructed, and the solution stability in Sobolev spaces was estimated. The problem of reconstructing a function from its integrals over an n-parameter family of conic surfaces with vertices ranging over a fixed coordinate axis was considered by S. V. Uspenskii.