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 A^–1 in a separable Hilbert space . For the approximate solution u_n of this equation by the least squares method in a coordinate system {e_k}_k that is an orthonormal basis of eigenvectors of a self-adjoint operator B similar to A ((B) = (A)), we give a priori estimates for the asymptotic behavior of the expressions r_n = u_n – u and R_n = Au_n – f as n . A relationship between the order of smallness of these expressions and the degree of smoothness of u with respect to the operator B is established.__________Translated from Funktsional nyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 85–90, 2005Original Russian Text Copyright © by M. L. GorbachukSupported by CRDF and Ukrainian Government Joint Grant UM1-2567-OD03.Translated by V. M. Volosov