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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Bibliographic Details
Published in:Journal of approximation theory Vol. 281-282
Main Author: Xiang, Jim X.
Format: Journal Article
Language:English
Published: Elsevier Inc 01.09.2022
Subjects:
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
Online Access:Get full text
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Summary: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