Approximation properties of λ-Bernstein operators

In this paper, we introduce a new type λ-Bernstein operators with parameter \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}λ∈[−1,1], we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Finally, we give some graphs and numerical examples to show the convergence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B_{n,\lambda }(f;x)$\end{document}Bn,λ(f;x) to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x)$\end{document}f(x), and we see that in some cases the errors are smaller than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B_{n}(f)$\end{document}Bn(f) to f.

In this paper, we introduce the new λ-Bernstein operators, B n,λ (f ; x) = n k=0b n,k (λ; x)f k n , ( 4 ) whereb n,k (λ; x) (k = 0, 1, . . . , n) are defined in (3) and λ ∈ [-1, 1]. This paper is organized as follows: In the following section, we estimate the moments and central moments of these operators (4). In Sect. 3, we investigate a Korovkin approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, and obtain a Voronovskaja-type asymptotic formula. In Sect. 4, we give some graphs and numerical examples to show the convergence of B n,λ (f ; x) to f (x) with different parameters.

Some preliminary results
Proof From (4), it is easy to prove n k=0b n,k (λ; x) = 1, then we can obtain (5). Next, as is well known, the Bernstein operators (1) preserve linear functions, that is to say, B n (at + b; x) = ax + b. We denote the latter two parts in the bracket of the last formula by 1 (n; x) and 2 (n; x), then we have B n,λ (t; x) = x + λ 1 (n; x) + 2 (n; x) .

Convergence properties
As we know, the space C We can obtain these three conditions easily by (5), (6) and (7) of Lemma 2.1. Thus the proof is completed.
where C is a positive constant, φ n (x) and ψ n (x) are defined in (16) and (17).
We have the following theorem.
Proof Since B n,λ (f ; x) are linear positive operators and f ∈ Lip M (α), we have Applying Hölder's inequality for sums, we obtain Thus, Theorem 3.4 is proved.
Finally, we give a Voronovskaja asymptotic formula for B n,λ (f ; x).
Proof Let x ∈ [0, 1] be fixed. By the Taylor formula, we may write where r(t; x) is the Peano form of the remainder, r(t; x) ∈ C[0, 1], using L'Hopital's rule, we have Applying B n,λ (f ; x) to (29), we obtain By the Cauchy-Schwarz inequality, we have B n,λ r(t; x)(tx) 2 ; x ≤ B n,λ r 2 (t; x); x B n,λ (tx) 4 Theorem 3.5 is proved.

Graphical and numerical analysis
In this section, we give several graphs and numerical examples to show the convergence of B n,λ (f ; x) to f (x) with different values of λ and n.
Let f (x) = 1 -cos(4e x ), the graphs of B n,-1 (f ; x) and B n,1 (f ; x) with different values of n are shown in Figs. 2 and 3. In Table 1, we give the errors of the approximation of B n,λ (f ; x) to f (x). We can see from Table 1 that in some special cases (such as n = 10, 20 and λ > 0), the errors of f -B n,λ (f ) ∞ are smaller than f -B n,0 (f ) ∞ (where B n,0 (f ; x) are classical Bernstein operators). Figure 4 shows the graphs of B n,λ (f ; x) with n = 10 and different values of λ.