Convergence of certain polynomial interpolants to a function defined on the unit circle
type: Item
instance of: scholarly article
title: Convergence of certain polynomial interpolants to a function defined on the unit circle
zbMATH Open document ID: 0675.30025
DOI: 10.1007/BF02170064
published in: Acta Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: Let f be analytic on the unit disk D and let \(f^{(r-1)}\) exist and be Riemann integrable on \(\partial D\). We denote by \(h_{rn-1}\) the polynomial of degree rn-1 which satisfies \[ h^{(j)}_{rn-1}(w_ k)=f^{(j)}(w_ k),\quad w_ k=e^{2\pi ik/n} \] for \(k=1,...,n\); \(j=0,...,r-1\). The authors show the compact convergence of \(h_{rn- 1}(z)\) to \[ (2\pi i)^{-1}\int_{| w| =1}f(w)/(w-z)dw \] in D. They obtain two corresponding results replacing \(h_{rn-1}\) by other polynomials which are characterized either by \(\ell_ 2\)-approximation or by the average of partial sums of \(h_{rn-1}\).
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 4105308
author: Alfred S. jun. Cavaretta
author: H. P. Dikshit
author: A. Sharma
reviewed by: Klaus Menke
MaRDI profile type: Publication
cites work: An extension of a theorem of Walsh
cites work: Extensions of a theorem of J. L. Walsh
cites work: On Walsh equiconvergence
cites work: Q3254471
full work available at URL: https://doi.org/10.1007/bf02170064
OpenAlex ID: W2089292234
description: scientific article; zbMATH DE number 4105308