Determining system poles using row sequences of orthogonal Hermite-Pad\'e approximants

TL;DR

This paper introduces a method to determine the poles of a system of functions using orthogonal Hermite-Padé approximants, providing convergence conditions and rates, and showing how denominators reveal pole locations.

Contribution

It develops a new approach linking orthogonal Hermite-Padé approximants with pole detection and convergence analysis for systems of functions.

Findings

Convergence of denominators is characterized under certain conditions.

Exact convergence rates of the denominators are established.

Denominators effectively detect the poles of the system functions.

Abstract

Given a system of functions $\textup{\textbf{F}}=(F_1,\ldots,F_d),$ 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 orthogonal expansions of the components $F_k, k=1,\ldots,d,$ with respect to a sequence of orthonormal polynomials associated with a measure $\mu$ whose support is contained in $E$. Such sequences of vector rational functions resemble row sequences of type II Hermite-Pad\'e approximants. Under appropriate assumptions on $\mu,$ 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 simultaneous approximants is estimated. It is shown that the common denominator of the approximants detect the location of the poles of the system of functions.