On the approximation of functions in [formula omitted] by a class of positive linear operators

Abstract

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.