Convergence rates of Padé and Padé-type approximants
type: Item
instance of: scholarly article
title: Convergence rates of Padé and Padé-type approximants
zbMATH Open document ID: 0856.41013
author: Amiran Ambroladze
author: Hans Wallin
published in: Journal of Approximation Theory
publication date: 1996-10-27T00:00:00Z
review text: Let \(f\) be holomorphic at \(\infty\) with \(f(\infty)= 0\), and let the polynomials \(p_n\), \(q_n\) and \(P_n\), \(Q_n\) of degree \(\leq n\) be so chosen that \[ q_n(z) f(z)- p_n(z)= {\mathcal O}(z^{- n- 1})\quad\text{and}\quad Q_n(z) f(z)- P_n(z)= {\mathcal O}(z^{- 1})\;(z\to \infty). \] In the first case, we get the Padé approximant (PA) \({p_n\over q_n}\) of order \(n\), in the second case, with \(Q_n\) a preassigned polynomial of degree \(n\), we get the Padé-type approximant (PTA) \({P_n\over Q_n}\). Actually, \(p_n\) and \(P_n\) are of degree \(\leq n- 1\). The authors show that if \(f\) is of the form \(f(z)= \int {w(t)\over z- t} d\mu(t)\), with an entire function \(w\) and a finite Borel measure on \(\mathbb{C}\) with compact support, the PTAs of order \(n\) give the same rate of convergence as the PAs of order \(n\). However, the PTAs are far easier to compute and need less information about \(f\). Theorem 1. If \(\mu\) is Lebesgue measure on \(\Delta= [- 1, 1]\), \(w\) an entire function, and \(Q_n\) are the Legendre polynomials, then \[ \limsup_{n\to \infty} \Biggl|f(z)- {P_n(z)\over Q_n(z)}\Biggr|^{1/n}\leq \exp(- 2g_\Omega(z)), \] locally uniformly on \(\Omega:= \mathbb{C}\backslash \Delta\), where \(g_\Omega\) is Green's function in \(\Omega\) with pole at \(\Omega\). Theorem 2 extends Theorem 1 to general measures, and Theorem 3 compares the errors \(R_n(z):= f(z)- {P_n(z)\over Q_n(z)}\), for a general entire \(w\), with those \(r_n(z)\) that are obtained when \(w(t)\equiv 1\). The convergence proofs depend on an integral representation of \(R_n(z)\) and a lemma for polynomial approximation of \({w(t)\over z- t}\).
Mathematics Subject Classification ID: 41A21
zbMATH DE Number: 938745
zbMATH Keywords: Padé approximation
zbMATH Keywords: Padé-type approximation
zbMATH Keywords: polynomial approximation
reviewed by: Dieter Gaier
MaRDI profile type: Publication
full work available at URL: jath.1996.0072
OpenAlex ID: W2014272708
DOI: 10.1006/JATH.1996.0072
description: scientific article; zbMATH DE number 938745