Some identities and inequalities for derivatives
type: Item
instance of: scholarly article
title: Some identities and inequalities for derivatives
zbMATH Open document ID: 0840.42002
published in: Journal of Approximation Theory
publication date: 1996-07-01T00:00:00Z
review text: The authors reprove inequalities due to Bernstein and Brudnyi, respectively, and prove a few other new inequalities involving the derivatives of trigonometric or algebraic polynomials. These proofs are based on the following identity due to \textit{M. Riesz} [Jahresber. Dtsch. Math.-Ver. 23, 354-368 (1914; J.-buch F.d.M. 45, 405)]: If \(\phi_m\) is a trigonometric polynomial of degree \(m\) and \(t_k:= (2k- 1)\pi/2m\), then \[ \phi_m'(x)= {1\over 4m} \sum^{2m}_{k= 1} \phi_m(x+ t_k)(- 1)^{k+ 1} (\sin{1\over 2} t_k)^{- 2}. \] Setting \(\phi_m(x):= (1/ m)\sin mx\) and \(x:= 0\) yields \(1= {1\over 4m^2} \sum^{2m}_{k= 1} (\sin{1\over 2} t_k)^{- 2}\).
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 26D05
zbMATH DE Number: 803693
zbMATH Keywords: inequalities
zbMATH Keywords: derivatives of trigonometric or algebraic polynomials
author: Katherine Balázs
author: Theodore Kilgore
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
full work available at URL: jath.1995.1078
OpenAlex ID: W2066057492
DOI: 10.1006/JATH.1995.1078
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description: scientific article; zbMATH DE number 803693