An extension of a theorem of Walsh
type: Item
instance of: scholarly article
title: An extension of a theorem of Walsh
zbMATH Open document ID: 0554.41015
DOI: 10.1007/BF03322500
author: H. P. Dikshit
author: A. Sharma
author: Alfred S. jun. Cavaretta
published in: Results in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Let \(f(z)\in A_{\rho}\) \((\rho >1)\), the class of function analytic in \(| z| <\rho\) but not in \(| z| \leq \rho\). For a given positive integer r, we study the polynomial \(P_{n,r}(z)\) which minimizes \(\sum^{r-1}_{\nu =0}\sum^{m-1}_{k=0}| Q_ n^{(\nu)}(\omega^ k)-f^{(\nu)}(\omega^ k)|^ 2,\) over all polynomials \(Q_ n\in \pi_ n\), where \(\omega^ m=1\), \(m=nq+c\). This leads to an extension of a recent result of \textit{T. J. Rivlin} [J. Approximation Theory 36, 334-345 (1982; Zbl 0504.42016)] on Walsh equiconvergence. Some other extensions and results related to those of Saff and Varga are given.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 3882952
zbMATH Keywords: Walsh equiconvergence
MaRDI profile type: Publication
cites work: Some remarks on a theorem of Saff and Varga
cites work: On Walsh equiconvergence
cites work: A note on the sharpness of J. L. Walsh's theorem and its extensions for interpolation in the roots of unity
cites work: Hermite-Birkhoff interpolation on roots of unity and Walsh equiconvergence
cites work: Approximation to a Function by a Polynomial in Another Function
full work available at URL: https://doi.org/10.1007/bf03322500
OpenAlex ID: W2007415322
description: scientific article; zbMATH DE number 3882952