On the convergence of type I Hermite-Pad\'e approximants

TL;DR

This paper establishes convergence theorems for type I Hermite-Padé approximants within Nikishin systems, filling a gap in the theoretical understanding of these vector rational approximations.

Contribution

It provides the first Markov and Stieltjes type convergence theorems for type I Hermite-Padé approximants, extending the theoretical framework.

Findings

Proves convergence of type I Hermite-Padé approximants for Nikishin systems.

Extends classical Markov and Stieltjes theorems to type I approximants.

Fills a theoretical gap in the convergence analysis of vector rational approximations.

Abstract

Pad\'e approximation has two natural extensions to vector rational approximation through the so called type I and type II Hermite-Pad\'e approximants. The convergence properties of type II Hermite-Pad\'e approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite-Pad\'e approximants for Nikishin systems of functions.