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

TL;DR

This paper establishes conditions for the geometric convergence of vector rational functions, constructed via simultaneous Padé-Faber approximants, and shows how their common denominators reveal the poles of the system closest to a given compact set.

Contribution

It provides necessary and sufficient conditions for convergence and explicitly characterizes the rate, extending the understanding of simultaneous Padé-Faber approximation.

Findings

Convergence conditions for the approximants are established.

The exact rate of convergence of denominators is determined.

Denominators detect the closest system poles and their order.

Abstract

Given a vector function ${\bf F}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the expansions of the components $F_k, k=1,\ldots,d,$ with respect to the sequence of Faber polynomials associated with $E$. Such sequences of vector rational functions are analogous to row sequences of type II Hermite-Pad\'e approximation. We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of the sequence of vector rational functions so constructed. The exact rate of convergence of these denominators is provided and the rate of convergence of the approximants is estimated. It is shown that the common denominators of the approximants detect the poles of the system of functions "closest" to $E$ and their order.