Sharp Jackson inequality with a special modulus of continuity
type: Item
instance of: scholarly article
title: Sharp Jackson inequality with a special modulus of continuity
zbMATH Open document ID: 1315.41004
DOI: 10.1134/S0081543814020047
author: A. G. Babenko
author: N. V. Dolmatova
author: Yu. V. Kryakin
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-12-17T00:00:00Z
review text: Let \(\mathbb{T}=\mathbb{R}/\mathbb{Z}=[-1/2,1/2)\) and \(C(\mathbb{T})\) be the the set of continuous \(1\)-periodic functions. Let \(E_{n-1}(f)\) be the best approximation of a function \(f\in C(\mathbb{T})\) by trigonometric polynomials of degree at most \(n-1\). For \(h>0\) denote by \(\chi_h\), the characteristic function of the interval \((-h/2,h/2)\), multiplied by the factor \((1/h)\). Denote by \(\ast\) the convolution product. The authors consider the second order modulus: \(W_2(f,\chi_h)=\| f\ast\chi_h\|=\left\| h^{-1} \int_0^{h/2}[f(\cdot-t)-2f(\cdot)+f(\cdot+t)]dt\right\|\), \(f\in C(\mathbb{T})\), \(h>0\). The main result is: \[ E_{n-1}(f)\leq\mathcal{F}_n(\alpha) W_2(f,\chi_{\alpha/(2n)}),\text{ for } f\in C(\mathbb{T}),\; n\in\mathbb{N},\;\alpha>2/\pi, \] where \(\mathcal{F}_n(\alpha)=\sec(\alpha^{-1})+\tan(\alpha^{-1})\). Moreover, the constant is the best possible for \(\alpha=2\nu+1\), \(\nu=0,1,\dots\) In order to obtain this result, the authors develop a technique based on the expansion \(f=f-f\ast \chi_h+\chi_h\ast(f-f\ast\chi_h)+\chi_h^2\ast(f-f\ast\chi_h)+\dots\) The modulus \(W_2(f,\chi_h)\) is compared with the \(K\)-functional \(K_2(f,h)=\inf_{g\in C^2}\{\| f-g\|+h^2\| D^2g\|\}\) by the inequalities \((1/4)K_2(f,h/(4\sqrt{6}))\leq W_2(f,\chi_h)\leq 4K_2(f,h/(4\sqrt{6}))\).
reviewed by: Radu Paltanea
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A44
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 6381062
zbMATH Keywords: direct theorems in approximation
zbMATH Keywords: best trigonometric approximation
zbMATH Keywords: second modulus of continuity
zbMATH Keywords: \(L\)-approximation
zbMATH Keywords: best constants
zbMATH Keywords: \(B\)-splines
zbMATH Keywords: degree of approximation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6381062