Incomplete Pad\'e approximation and convergence of row sequences of Hermite-Pad\'e approximants

TL;DR

This paper extends classical convergence theorems for Hermite-Padé approximants by introducing incomplete Padé approximation, providing refined results and broader applicability in the analysis of vector-valued analytic functions.

Contribution

It introduces the concept of incomplete Padé approximation and proves a Montessus de Ballore type theorem for row sequences of Hermite-Padé approximants.

Findings

Refined convergence results for Hermite-Padé approximants

Introduction of incomplete Padé approximation concept

Broader applicability to systems of approximants

Abstract

We give a Montessus de Ballore type theorem for row sequences of Hermite-Pad\'e 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 Pad\'e approximation which contains, in particular, simultaneous Pad\'e approximation and may be applied in the study of other systems of approximants as well.