Direct and Inverse Results for Multipoint Hermite-Pade Approximants

TL;DR

This paper establishes necessary and sufficient conditions for the geometric convergence of multipoint Hermite-Pade approximants, linking convergence rates to the functions' analytic properties and pole locations.

Contribution

It provides a comprehensive characterization of convergence behavior and pole detection for multipoint Hermite-Pade approximants based on analytic function properties.

Findings

Necessary and sufficient conditions for convergence are identified.

Exact convergence rates are derived in relation to function analyticity.

Pole locations closest to the domain are detectable through these results.

Abstract

Given a system of functions f = (f1, . . . , fd) analytic on a neighborhood of some compact subset E of the complex plane, we give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of multipoint Hermite-Pade approximants. The exact rate of convergence of these denominators and of the approximants themselves is given in terms of the analytic properties of the system of functions. These results allow to detect the location of the poles of the system of functions which are in some sense closest to E.