Abstract
In this paper, we introduce a family of GBS operators of bivariate tensor product of λ-Bernstein–Kantorovich type. We estimate the rate of convergence of such operators for B-continuous and B-differentiable functions by using the mixed modulus of smoothness, establish the Voronovskaja type asymptotic formula for the bivariate λ-Bernstein–Kantorovich operators, as well as give some examples and their graphs to show the effect of convergence.
Highlights
In 1912, Bernstein [1] constructed a sequence of polynomials to prove the Weierstrass approximation theorem as follows: nkBn(f ; x) = f n bn,k(x), (1)k=0 for any continuous function f ∈ C[0, 1], where x ∈ [0, 1], n = 1, 2, . . . , and Bernstein basis functions bn,k(x) are defined by bn,k(x) =n k xk(1 – x)n–k. (2)The polynomials in (1), called Bernstein polynomials, possess many remarkable properties.Recently, Cai et al [2] proposed a new type λ-Bernstein operators with parameter λ ∈ [–1, 1], they obtained some approximation properties and gave some graphs and numerical examples to show that these operators converge to continuous functions f
We use the mixed modulus of smoothness to estimate the rate of convergence of GBS operators of bivariate tensor product of λBernstein–Kantorovich type for B-continuous and B-differentiable functions, and establish a Voronovskaja type asymptotic formula for the bivariate λ-Bernstein–Kantorovich operators
This paper is mainly organized as follows: In Sect. 2, we introduce the bivariate tensor product of λ-Bernstein–Kantorovich operators Kmλ1,n,λ2 (f ; x, y) and the GBS operators UKmλ1,n,λ2 (f ; x, y)
Summary
In 1912, Bernstein [1] constructed a sequence of polynomials to prove the Weierstrass approximation theorem as follows: nk. Cai et al [2] proposed a new type λ-Bernstein operators with parameter λ ∈ [–1, 1], they obtained some approximation properties and gave some graphs and numerical examples to show that these operators converge to continuous functions f. We use the mixed modulus of smoothness to estimate the rate of convergence of GBS operators of bivariate tensor product of λBernstein–Kantorovich type for B-continuous and B-differentiable functions, and establish a Voronovskaja type asymptotic formula for the bivariate λ-Bernstein–Kantorovich operators. 4, the rate of convergence for B-continuous and B-differentiable functions of GBS operators UKmλ1,n,λ2 (f ; x, y) is proved. The GBS operators of the bivariate tensor product of λ-Bernstein–Kantorovich type are defined as UKmλ1,n,λ2 f (t, s); x, y. UK110,1,10(f (s, t); x, y) in Fig. 4 to compare the bivariate λ-Bernstein–Kantorovich operators with GBS operators
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