Abstract
In this paper, we introduce (p,q)-gamma operators which preserve x^{2}, we estimate the moments of these operators, and establish direct and local approximation theorems of these operators. Then two approximation theorems about Lipschitz functions are obtained. The estimates on the rate of convergence and some weighted approximation theorems of the operators are also obtained. Furthermore, the Voronovskaja-type asymptotic formula is also presented.
Highlights
1 Introduction With the rapid development of the approximation theory about the operators since the last century, lots of operators, such as Bernstein operators [4], Szász–Mirakjan operators [32, 37], Baskakov operators [3], Bleimann–Butzer–Hann operators [5], and Meyer–König– Zeller operators [31], have been proposed and constructed by several researchers due to Weierstrass and the important convergence theorem of Korovkin [26], see [17]
With the rapid development of q-calculus [22], the study of new polynomials and operators constructed with q-integer has attracted more and more attention
The fourth theorem is a result about the rate of convergence for the operators Gnp,q(f ; x): Theorem 3.4 Let f ∈ Cx2 [0, ∞), 0 < q < p ≤ 1, and a > 0, we have
Summary
With the rapid development of the approximation theory about the operators since the last century, lots of operators, such as Bernstein operators [4], Szász–Mirakjan operators [32, 37], Baskakov operators [3], Bleimann–Butzer–Hann operators [5], and Meyer–König– Zeller operators [31], have been proposed and constructed by several researchers due to Weierstrass and the important convergence theorem of Korovkin [26], see [17]. Let δ > 0 and CB2[0, ∞) = {g : g , g ∈ CB[0, ∞)}, the following K -functional is defined: K(f ; δ) = inf f – g + δ g . Let F be a subset of the interval [0, ∞), we define that f ∈ LipM(γ , F) if the following inequality f (x) – f (y) ≤ M|x – y|γ , x ∈ F and y ∈ [0, ∞)
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