Converse results on equiconvergence of interpolating polynomials
type: Item
instance of: scholarly article
title: Converse results on equiconvergence of interpolating polynomials
zbMATH Open document ID: 0666.30023
DOI: 10.1007/BF02020764
published in: Analysis Mathematica
publication date: 1988-01-01T00:00:00Z
review text: Let \(U_ n:=\{e^{2\pi ik/n}\), \(k=0,...,n-1\}\), \(U=\cup^{\infty}_{n=1}U_ n\) be the roots of unity and \((a_ k)\) be a null sequence of complex numbers. Let f be defined on U and \[ \Delta_{\ell,n-1}(z):=L_{n-1}(z)-\sum^{n-1}_{k=0}\sum^{\ell - 1}_{j=0}a_{k+jn}z^ k,\quad \ell \in {\mathbb{N}}, \] where \(L_{n-1}\) is the Lagrange interpolant to f on \(U_ n\). The authors \(prove:\) (\(\Delta_{\ell,n-1})\) is uniformly bounded on compact subsets of \(| z| <\rho^{1+\ell}\) if and only if \(g(z):=\sum^{\infty}_{k=0}a_ kz^ k\) is analytic in \(| z| <\rho\) and there exists a function h analytic in \(| z| <\rho^{1+\ell}\) such that \(g(z)+h(z)=f(z)\) for \(z\in U\). This theorem contains a result of \textit{J. Szabados} [Analysis 2, 267-280 (1982; Zbl 0532.30002)].
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 4089865
author: Kamen Ivanov
author: A. Sharma
reviewed by: Klaus Menke
MaRDI profile type: Publication
cites work: Interpolation in the roots of unity: An extension of a theorem of J. L. Walsh
cites work: Converse results in the Walsh theory of overconvergence
cites work: CONVERSE RESULTS IN THE THEORY OF OVERCONVERGENCE OF COMPLEX INTERPOLATING POLYNOMIALS
cites work: A note on the sharpness of J. L. Walsh's theorem and its extensions for interpolation in the roots of unity
full work available at URL: https://doi.org/10.1007/bf02020764
OpenAlex ID: W123575796
description: scientific article; zbMATH DE number 4089865