The optimal set of the modulus of continuity in the sharp Jackson inequality in the space \(L_ 2\).
type: Item
instance of: scholarly article
title: The optimal set of the modulus of continuity in the sharp Jackson inequality in the space \(L_ 2\).
zbMATH Open document ID: 1070.41004
author: Elena E. Berdysheva
published in: Mathematical Notes
publication date: 2005-06-21T00:00:00Z
review text: To a function \(f\in L_2[-\pi,\pi]\) and a compact set \(Q\subset [-\pi,\pi]\) is denoted the supremum \(w(f,\varphi)=\sup_{t\in Q}\| f(\cdot,+t)-f(\cdot)\|)_{L_2[-\pi,\pi]}\) which is an analog of the modules of continuity. If is denoted by \(K(n,Q)\) the best constant in Jackson's inequality between the best approximation of the function \(f\) by trigonometric polynomials of degree \((n-1)\) in the space \(L_2[-\pi,\pi]\) and the modulus of continuity \(w(f,Q)\) on the strength of a result of Yudin, it is shown that if the measure of the set \(Q\) is less than \(\underline\pi\) then \(K(n,Q)>\frac1{\sqrt 2}\). It was known from results of Chernykh that \(K(n,Q)\geq \frac1{\sqrt2}\) and \(K[n,[0,\frac\pi n]]=\frac1{\sqrt2}\). The strict inequality proved that \(K(n,Q)\geq \frac1{\sqrt2}\) and \(K[n,[0,\frac\pi n]]=\frac1{\sqrt2}\). The strict inequality proved in this paper means that \(K(n,Q)\) as function of the measure of \(Q\) have a strict maximum when the measure of \(Q\) is \(\frac \pi n\) in \([0,\underline\pi]\).
reviewed by: Francisco Pérez Acosta
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 2177674
zbMATH Keywords: Jackson's inequality
zbMATH Keywords: modulus of continuity
zbMATH Keywords: approximation by trigonometric polynomials
MaRDI profile type: Publication
full work available at URL: b:matn.0000049661.88696.b3
OpenAlex ID: W2019537561
DOI: 10.1023/B:MATN.0000049661.88696.B3
description: scientific article; zbMATH DE number 2177674