Direct and inverse results on row sequences of Hermite-Pad\'e approximants

TL;DR

This paper establishes necessary and sufficient conditions for the geometric convergence of Hermite-Padé approximants' denominators, providing explicit convergence rates based on intrinsic properties of the function system.

Contribution

It offers a complete characterization of convergence conditions and explicit rates for Hermite-Padé approximants, advancing understanding of their approximation behavior.

Findings

Conditions for geometric convergence are characterized precisely.

Explicit convergence rates are derived for the approximants.

Results depend on intrinsic properties of the function system.

Abstract

We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.