CONVERGENCE THEOREMS FOR ROWS OF HERMITE-PADE INTEGRAL APPROXIMANTS

The object of this paper is the study of the convergence of integral approximants, which are a special case of Hermite-Pade approximants of Latin type, to functions which are analytic in a disk except for one interior singular point. We give detailed estimates of the rate of convergence of the sequence of approximants of type [L/M; 1] for fixed M, as L --> infinity, in a model case study. We also give estimates of the rate of convergence of approximants of type [L/M; 1; 2] for fixed M, as L --> infinity, for a model exhibiting a confluent singularity. We prove that integral approximants of these types converge uniformly on compact subsets of the disk which is centered on the origin and has the singular point of the given function on its boundary. We further prove convergence on additional Riemann sheets beyond the principal one in a lune near the singular point.