On an open problem of P. Turán concerning Birkhoff interpolation based on the roots of unity
type: Item
instance of: scholarly article
title: On an open problem of P. Turán concerning Birkhoff interpolation based on the roots of unity
zbMATH Open document ID: 0598.41003
DOI: 10.1016/0021-9045(86)90034-1
author: A. K. Varma
author: Jószef Szabados
published in: Journal of Approximation Theory
publication date: 1986-01-01T00:00:00Z
review text: Let f(z) be analytic in \((z)<1\) and continuous in (z)\(\leq 1\) and let \(z_{kn}\) \((k=1,...,n)\) denote the nth roots of unity. Let \(R_{n,m}(f;z):=\sum^{n}_{k=1}f(z_{kn})A_{k,n,m}(z)+\beta_{k,n, m}B_{k,n,m}(z)\) where \(A_{k,n,m}(z)\), \(B_{k,n,m}(z)\) are the fundamental polynomials of (0,m) interpolation on \(\{z_{kn}\}^ n_ 1\) and \(\beta_{k,n,m}=0(n^ m/\ell n n)\), \(k=1,...,n\). The main result of the authors is that \[ \lim_{n\to \infty}\frac{1}{2\pi}\int_{(z)=1}| f(z)-R_{n,m}(f;z)|^ 2| dz| =0 \] provided \(\beta_{kn}=0(n^ m)\), \(n=1,...,n\). This solves a problem proposed by \textit{P. Turán} in 1974 [J. Approximation Theory 29, 23-89 (1980; Zbl 0454.41001)] for the special case \(m=2\).
reviewed by: A. Sharma
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 3964599
zbMATH Keywords: fundamental polynomials
MaRDI profile type: Publication
full work available at URL: 0021-9045(86)90034-1
OpenAlex ID: W1967917613
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description: scientific article; zbMATH DE number 3964599