Some inequalities for derivatives of trigonometric and algebraic polynomials
type: Item
instance of: scholarly article
title: Some inequalities for derivatives of trigonometric and algebraic polynomials
zbMATH Open document ID: 0861.26006
DOI: 10.1007/BF03322182
author: Michael Felten
author: Theodore Kilgore
published in: Results in Mathematics
publication date: 1997-04-29T00:00:00Z
review text: Let \(W\) be a \(2\pi\)-periodic even weight of class \(A_p\) and \(T_n:\mathbb{R}\to \mathbb{R}\) a trigonometric polynomial of degree at most \(n\). Then for all \(\ell\in \mathbb{N}\), \[ |T_n^{(\ell)} |_{p,W}\leq 3\ell(2n)^\ell \omega \bigl(T_n; {\textstyle {\pi\over4n}} \bigr)_{p,W}. \] Here \(|.|_{p,W}\) stands for the norm in the weighted Lebesgue space \(L^p_W((0,2\pi))\) and the modulus of continuity \(\omega(T_n;\delta)\), \(\delta\geq 0\), is given by \[ \omega(T_n; \delta)_{p,W}= \sup_{|h-k|\leq \delta}|T_n(.+h)- T_n(.+k) |_{p,W}. \] This is one of the main results of the paper which generalizes to higher derivatives Theorem 1 of \textit{K. Balázs} and \textit{Th. Kolgore} [J. Approximation Theory 82, 274-286 (1995; Zbl 0840.42002)]. Applications to algebraic polynomials are given. The authors also prove an inequality of Brudnyi in terms of \(r\)th order moduli of continuity \(\omega_r\).
Mathematics Subject Classification ID: 26D05
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 42A05
zbMATH DE Number: 933258
zbMATH Keywords: trigonometric polynomials
zbMATH Keywords: inequalities for derivatives
zbMATH Keywords: weighted Lebesgue space
zbMATH Keywords: modulus of continuity
zbMATH Keywords: algebraic polynomials
reviewed by: Bohumír Opic
MaRDI profile type: Publication
cites work: Some identities and inequalities for derivatives
cites work: On weighted \(L^p\) simultaneous approximation
cites work: Hardy's inequality with weights
cites work: Weighted Norm Inequalities for the Hardy Maximal Function
cites work: Q3997067
full work available at URL: https://doi.org/10.1007/bf03322182
OpenAlex ID: W2086931954
description: scientific article; zbMATH DE number 933258