Approximation by Boolean sums of positive linear operators. II: Gopengauz-type estimates
type: Item
instance of: scholarly article
title: Approximation by Boolean sums of positive linear operators. II: Gopengauz-type estimates
zbMATH Open document ID: 0732.41022
DOI: 10.1016/0021-9045(89)90085-3
author: Jiading Cao
author: Heiner Gonska
published in: Journal of Approximation Theory
publication date: 1989-01-01T00:00:00Z
review text: A theorem of Gopengauz says that for every \(f\in C[-1,1]\) there exists an algebraic polynomial \(p_ n\) of degree n (depending on f) such that there holds \[ | f(x)-p_ n(x)| \leq c\omega_ 2(f,\sqrt{1-x^ 2}\cdot n^{-1}),\quad | x| \leq 1, \] where \(\omega_ 2\) denotes the classical second order modulus of continuity and c is an absolute constant independent of n, f and x. The main result of the present paper gives a construction for algebraic polynomials of degree sn-s, \(s\geq 3\), furnishing an estimate of the same type. This construction is based on the Boolean sum of positive linear and polynomial operators introduced by Pichugov and Lehnhoff respectively and certain interpolation operators. [For part I see Rend. Mat. Appl. VII., Ser. 6, No.4, 525-546 (1986; Zbl 0689.41019).]
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 4210762
zbMATH Keywords: Gopengauz estimates
zbMATH Keywords: algebraic polynomial
zbMATH Keywords: Boolean sum
reviewed by: Manfred Wolfgang Müller
MaRDI profile type: Publication
full work available at URL: 0021-9045(89)90085-3
OpenAlex ID: W2017492192
cites work: Polynomial Interpolation to Boundary Data on Triangles
cites work: Q3031416
cites work: Pointwise approximation by algebraic polynomials
cites work: The approximation of continuous functions by positive linear operators
cites work: Q4751122
cites work: Q4112374
cites work: Q3256418
cites work: Über die Approximation reeller stetigen Funktionen durch gewöhnliche Polynome
cites work: Q3472614
cites work: Q4040503
cites work: Q3752029
cites work: Punktweise Abschätzungen zur Approximation durch algebraische Polynome. (Pointwise estimates for approximation by algebraic polynomials)
cites work: Q5577334
cites work: A simple proof of A. F. Timan's theorem
cites work: A new proof of Teljakowskii's theorem
cites work: Q3785328
cites work: On the approximation of functions by some singular integrals
cites work: Q4179343
cites work: Q5522339
cites work: Q3778409
description: scientific article; zbMATH DE number 4210762