Rational best approximants in the Hardy space \(H^2\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 0989.41009
publication date: 2002-07-22T00:00:00Z
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 1721610
zbMATH Keywords: multipoint Padé approximants
zbMATH Keywords: rational best approximation in the \(H^2\) norm
zbMATH Keywords: uniqueness
zbMATH Keywords: asymptotic error estimates
zbMATH Keywords: Markov functions
author: Herbert Stahl
MaRDI profile type: Publication
title: Rational best approximants in the Hardy space \(H^2\)
review text: In this survey paper the author presents (without proofs) recent developments in the study of rational best approximation of Markov functions in the norm of the Hardy space \(H^2(V)\) with \(V=\{z\in {\overline{\mathbb{C}}}:|z|>1\}.\) The author has a major contribution in this field. We recall that a Markov function \(f\) with respect to positive measure \(\mu,\) \((supp(\mu)\subseteq (-1,1))\) is represented by \(f(z)=\int(t-z)^{-1}d\mu(t).\) The analysis of rational best approximants in \(H^2(V)\) requires some results about orthogonal polynomials with varying weights and of multipoint Padé approximants. Then, asymptotic error estimates in the weak and strong sense for multipoint Padé approximants are presented at the beginning. NEWLINENEWLINENEWLINELet \(\mathcal{R}^1_{n-1,n}\) be the family of all rational functions in \(H^2(V),\) of the form \(p_{n-1}/p_n,\) with poles in the unit disk, \(p_k\) being a polynomial of degree \(k.\) Let \(R^*_n\in {\mathcal R}^1_{n-1,n}\) be a best approximant of the Markov function \(f.\) The author gives error estimates of \(\|f-R^*_n\|^{1/n}_{H^2}\) and \(\|f-R^*_n\|_{H^2},\) as well as the point-wise variants, for \(n\rightarrow\infty.\) The estimates are expressed in terms of the condenser capacity for a pair of disjoint compact sets. The uniqueness of \(R^*_n\) is obtained for large \(n.\)NEWLINENEWLINEFor the entire collection see [Zbl 0971.00015].
reviewed by: Ioan Şerb
description: scientific article; zbMATH DE number 1721610