On certain mean values of polynomials on the unit interval.
type: Item
instance of: scholarly article
title: On certain mean values of polynomials on the unit interval.
zbMATH Open document ID: 1063.41008
published in: Journal of Approximation Theory
publication date: 1999-01-01T00:00:00Z
review text: For any continuous function \(f: [-1,1] \rightarrow {\mathbb C}\) and any \(p \in (0,\infty)\), let \(\| f \|_p\) and \(\| f\|_\infty\) denote the \(L_p\) and the sup norm of \(f\) respectively. It is known [\textit{S. M. Nikolskij}, Tr. Mat. Inst. Steklova 38, 244--278 (1951; Zbl 0049.32301)] that if \(f\) is a polynomial of degree \(n\), then for all \(p>0\), \(\| f \|_\infty \leq C_p n^{2/p} \| f \|_p\) where \(C_p\) is a constant dependent on \(p\) but not on \(n\). Here the order of the magnitude of the bound is the best possible. The authors obtain a sharp version of the foregoing inequality for polynomials not vanishing in the open unit disk. As an application, an extension of an inequality of \textit{P. Erdős} [Ann. Math. (2) 41, 310--313 (1940; Zbl 0024.00403 and JFM 66.0303.01)] is obtained.
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 1389112
zbMATH Keywords: Mean values
zbMATH Keywords: polynomials
zbMATH Keywords: unit interval
author: Dimiter P. Dryanov
author: Qazi I. Rahman
reviewed by: H. P. Dikshit
MaRDI profile type: Publication
full work available at URL: jath.1999.3364
OpenAlex ID: W2062649376
cites work: Q5535091
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DOI: 10.1006/JATH.1999.3364
description: scientific article; zbMATH DE number 1389112