On the approximation of functions in C1(I) by a class of positive linear operators

Let I be a finite or infinite interval and ω(f′,δ)=sup|x−y|≤δ|f′(x)−f′(y)| be the modulus of continuity of f′ for f(x)∈C1(I)={f:f′∈C(I)}. Let [a,b]⊂I be a finite close interval. In this paper we derive the best asymptotic constant defined by C=lim supn→∞supf∈C1(I)supx∈[a,b]n|Ln(f,x)−f(x)|ω(f′,1n)whe...

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Vydáno v:	Journal of approximation theory Ročník 281-282
Hlavní autor:	Xiang, Jim X.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Elsevier Inc 01.09.2022
Témata:	B-spline operator Baskakov operator Bernstein operator Best asymptotic constant Gamma operator Modulus of continuity Szász–Mirakyan operator Bernstein operator Best asymptotic constant Baskakov operator Gamma operator B-spline operator Szász–Mirakyan operator Modulus of continuity
ISSN:	0021-9045, 1096-0430
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Popis
Shrnutí:	Let I be a finite or infinite interval and ω(f′,δ)=sup\|x−y\|≤δ\|f′(x)−f′(y)\| be the modulus of continuity of f′ for f(x)∈C1(I)={f:f′∈C(I)}. Let [a,b]⊂I be a finite close interval. In this paper we derive the best asymptotic constant defined by C=lim supn→∞supf∈C1(I)supx∈[a,b]n\|Ln(f,x)−f(x)\|ω(f′,1n)where f is not a linear function of x and Ln(f,x) are a class of positive linear operators. The results are applied to some well-known operators including Szász–Mirakyan operator, Gamma operator, Baskakov operator and B-spline operator.
ISSN:	0021-9045 1096-0430
DOI:	10.1016/j.jat.2022.105800