An algorithm with optimal convergence rate for solving Fredholm equations of the first kind

The problem of finite-dimensional approximation for some classes of Fredholm equations of the first kind is considered in the case when the kernel and the right-hand side are given not exactly. An algorithm achieving an optimal order of accuracy for the recovery of normal solutions is proposed. This algorithm is based on the nonstationary iterated Tikhonov method, the generalized residual principle, and a multi-projection scheme of discretization. It is found that using this method leads to an required accuracy of approximation at economic expenses of discrete information in the form of Fourier-Legendre coefficients. The efficiency of numerical realization of the proposed algorithm is confirmed by a model example.