Incomplete Padé approximation and convergence of row sequences of Hermite–Padé approximants

We give a Montessus de Ballore type theorem for row sequences of Hermite–Padé approximations of vector valued analytic functions refining some results in this direction due to P.R. Graves-Morris and E.B. Saff. We do this introducing the notion of incomplete Padé approximation which contains, in particular, simultaneous Padé approximation and may be applied in the study of other systems of approximants as well.