# Determining system poles using row sequences of orthogonal Hermite-Pade approximants Articles

## Overview

### published in

- JOURNAL OF APPROXIMATION THEORY Journal

### publication date

- July 2018

### start page

- 15

### end page

- 40

### volume

- 231

### Digital Object Identifier (DOI)

### International Standard Serial Number (ISSN)

- 0021-9045

### Electronic International Standard Serial Number (EISSN)

- 1096-0430

### abstract

- Given a system of functions F = (F-1,..., F-d), analytic on a neighborhood of some compact subset E of the complex plane with simply connected complement in the extended complex plane, we define a sequence of vector rational functions with common denominator in terms of the orthogonal expansions of the components F-i, i = 1,.., 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-Pade 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 denominators of the approximants detect the location of the poles of the system of functions. (C) 2018 Elsevier Inc. All rights reserved.

### keywords

- montessus de ballore theorem; orthogonal expansions; simultaneous approximation; hermite-pade approximation; rate of convergence; inverse results