Application of the Lagrange principle in the problem of optimal inversion of linear operators in the case of sourcewise representability of the exact solutions to operator equations

The problem of approximate solution of a linear operator equation with a priori information on its solution is studied. A problem whose exact solution belongs to a ball image is considered when a linear continuous operator is used for mapping. The problem of optimal recovery of linear continuous functionals is solved with the use of a priori information. An algorithm for finding a method and an error of optimal recovery is described.