On the nonclassical approximation method for periodic functions by trigonometric polynomials
type: Item
instance of: scholarly article
title: On the nonclassical approximation method for periodic functions by trigonometric polynomials
zbMATH Open document ID: 1275.41023
author: Yu. S. Kolomoitsev
author: R. M. Trigub
published in: Journal of Mathematical Sciences (New York)
publication date: 2013-04-09T00:00:00Z
review text: A periodic function is approximated in the uniform norm by the linear polynomial means of its Fourier series. The means in question are generated by a multiplier function \(\varphi(x)\) supported in \([-\pi,\pi]\) and such that not only \(\varphi(0)=1\) but, contrary to the classical situation, there are other points at which \(\varphi\) takes the value \(1\). Three theorems are proven in the paper. They have something in common, more precisely, in each of them a necessary and sufficient condition for the rate of approximation to be equivalent either to a certain modulus of smoothness or to a \(K\)-functional is given in terms of the belonging of the ratio of \(1-\varphi(x)\) and a certain function, related to the points where \(\varphi\) is equal to \(1\), to the Wiener algebra of absolutely convergent Fourier integrals. Unfortunately, the paper is not well enough translated, with numerous deviations from standard language and even statements that are misleading in translation.
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A45
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 42A45
zbMATH DE Number: 6152303
zbMATH Keywords: Fourier series
zbMATH Keywords: Fourier transform of a measure
zbMATH Keywords: multiplier
zbMATH Keywords: comparison principle
zbMATH Keywords: K-functional
zbMATH Keywords: moduli of smoothness
zbMATH Keywords: Wiener's theorem
MaRDI profile type: Publication
full work available at URL: s10958-012-1111-x
OpenAlex ID: W2056449229
cites work: Exact estimates of the rate of approximation of convolution operators
cites work: Q3858736
cites work: Q4843130
cites work: The Wiener algebra of absolutely convergent Fourier integrals: an overview
cites work: Q5640160
cites work: Q3909485
cites work: Q3254471
DOI: 10.1007/S10958-012-1111-X
description: scientific article; zbMATH DE number 6152303