The Bézier variant of Kantorovich type λ-Bernstein operators

In this paper, we introduce the Bézier variant of Kantorovich type λ-Bernstein operators with parameter λ∈[−1,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda\in[-1,1]$\end{document}. We establish a global approximation theorem in terms of second order modulus of continuity and a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness. Finally, we combine the Bojanic–Cheng decomposition method with some analysis techniques to derive an asymptotic estimate on the rate of convergence for some absolutely continuous functions.

In this paper, we propose the Kantorovich type λ-Bernstein operators and the Bézier variant of Kantorovich type λ-Bernstein operators where 1], and λ ∈ [-1, 1]. Obviously, when α = 1, L n,λ,1 (f ; x) reduce to Kantorovich type λ-Bernstein operators (5); when λ = 0, L n,0,α (f ; x) reduce to Bernstein-Kantorovich-Bézier operators defined in [13]; when λ = 0, α = 1, L n,0,1 (f ; x) reduce to Bernstein-Kantorovich operators defined in [13]. Let where χ k (t) is the characteristic function on the interval [ k n+1 , k+1 n+1 ] with respect to [0, 1]. By the Lebesgue-Stieltjes integral representations, we have The aims of this paper are to study the rate of convergence of operators L n,λ,α for f ∈ C [0,1] and the asymptotic behavior of L n,λ,α for some absolutely continuous functions f ∈ DB , where the class of functions of DB is defined by For a bounded function f on [0, 1], the following metric forms were first introduced in [12]: For the basic properties of , and x (f ; μ), refer to [12].

Some lemmas
For proving the main results, we need the following lemmas.

Main results
As we know, the space [16], there exists an absolute constant C > 0 such that where ω 2 (f ; t) := sup 0<h≤t sup x,x+h,x+2h∈ [0,1] |f (x + 2h) -2f (x + h) + f (x)| is the second order modulus of smoothness of f ∈ C [0,1] . We also denote the usual modulus of continuity of where C is a positive constant.
Proof Let g ∈ C 2 [0,1] , by Taylor's expansion As we know, L n,λ,α (1; x) = 1. Applying L n,λ,α (·; x) to both sides of the above equation, we get By the Cauchy-Schwarz inequality, (12) and Lemma 2.4, we have Then, using the above inequality, we have Hence, taking infimum on the right-hand side over all g ∈ C 2 [0,1] , we get By (15), we obtain This completes the proof of Theorem 3.1.
Next, we recall some definitions of the Ditzian-Totik first order modulus of smoothness and K -functional, which can be found in [17]. Let f ∈ C [0,1] , and ϕ(x) := √ x(1x), the first order modulus of smoothness is given by 1] is the class of all absolutely continuous functions on [0, 1]. Besides, from [17], there exists a constant C > 0 such that where C is a positive constant.
Proof Since applying L n,λ,α (f ; x) to the above equality, we have We will estimate t x g (u) du: For any x, t ∈ (0, 1), we have From (18), using the Cauchy-Schwarz inequality, we obtain Hence, using the above inequality, we have Taking infimum on the right-hand side over all g ∈ C ϕ [0,1] , we get By (17), we obtain Theorem 3.2 is proved.
Finally, we study the approximation properties of L n,λ,α (f ; x) for some absolutely continuous functions f ∈ DB .

Theorem 3.3 Let f be a function in DB .
If φ(x+) and φ(x-) exist at a fixed point x ∈ (0, 1), then we have where [n] denotes the greatest integer not exceeding n, and Proof By the fact that L n,λ,α (1; x) = 1, using (7) and (8), we have By the Bojanic-Cheng decomposition [18], we have where φ x (u) is defined in (19), sgn(u) is a sign function and δ x (u) = 1, u = x; 0, u = x. By direct integrations, we find that where Integration by parts derives Note that R n,λ,α (x, t) ≤ 1 and φ x (x) = 0, it follows that x φ x ; x k .
From Lemma 2.5 (i) and change of variable t = xx/u, we have x φ x ; x k .

Conclusion
In this paper, we have presented a Bézier variant of Kantorovich type λ-Bernstein operators L n,λ,α (f ; x), and established approximation theorems by using the usual second order modulus of smoothness and the Ditzian-Totik modulus of smoothness. From Theorem 3.3 of Sect. 3, we know that the rate of convergence of operators L n,λ,α (f ; x) for f ∈ DB is 1 √ n+1 . Furthermore, we might consider the approximation of these operators L n,λ,α (f ; x) for locally bounded functions.