Riemann-Hilbert analysis and uniform convergence of rational interpolants to the exponential function
type: Item
instance of: scholarly article
title: Riemann-Hilbert analysis and uniform convergence of rational interpolants to the exponential function
zbMATH Open document ID: 1069.30062
published in: Journal of Approximation Theory
publication date: 2005-02-22T00:00:00Z
review text: The polynomials \(p\) and \(q\) of degrees \(n\) constituting rational approximants to the exponential function satisfy the relation \(p(z)\exp(-z/2)+q(z)\exp(z/2)= O(\omega_{2n+1}(z))\), as \(z\) tends to the roots of the complex polynomial \(\omega_{2n+1}(z)\). The large \(n\) asymptotics is studied of the polynomials \(p\) and \(q\), which are characterized by a Riemann-Hilbert problem for a \(2\times2\) matrix valued function. The Deift-Zhou steepest descent method for Riemann-Hilbert problems is used to obtain strong uniform asymptotics for the scaled polynomials \(p(2nz)\) and \(q(2nz)\) in every domain of the complex plane. A precise estimate of the error function is given.
reviewed by: Nico M. Temme
Mathematics Subject Classification ID: 30E15
Mathematics Subject Classification ID: 33B10
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 2136998
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: exponential function
zbMATH Keywords: rational interpolation
zbMATH Keywords: uniform convergence
zbMATH Keywords: nonlinear steepest descent
author: F. Wielonsky
MaRDI profile type: Publication
full work available at URL: j.jat.2004.07.004
OpenAlex ID: W2059492457
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DOI: 10.1016/J.JAT.2004.07.004
description: scientific article; zbMATH DE number 2136998