On the approximation to solutions of operator equations by the least squares method
We consider the equation Au = f, where A is a linear operator with compact inverse in a Hilbert space. For the approximate solution u_n of this equation by the least squares method in a coordinate system that is an orthonormal basis of eigenvectors of a self-adjoint operator B similar to A (D (A) = D (B)), we give a priori estimates for the asymptotic behavior of the expression R_n = \|Au_n - f\| as n → ∞. A relationship between the order of smallness of this expression and the degree of smoothness of the solution u with respect to the operator B (direct and converse theorems) is established., number=3, journal=Methods of Functional Analysis and Topology, author=Gorbachuk, M. L. and Gorbachuk, V. I., year=2008, month=Sep., pages=229–241