The Lagrange principle and finite-dimensional approximations in the optimal inverse problem for linear operators

This paper is devoted to the Lagrange principle for optimal recovery in the problem of solving operator equations. Some optimal recovery problems and a more general problem are formulated. The relation between the problem in infinite-dimensional space and its analogue in finite-dimensional space is studied. A theorem on common optimal recovery methods for the problems in infinite-dimensional space and in finite-dimensional space is proved. The problem in infinite-dimensional space is approximated by problems in finite-dimensional spaces. A new optimal method for the problem of solving operator equations in finite-dimensional space is described. This problem is considered as a system of linear algebraic equations with a priori information on its solution.