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In this note, we study some approximation properties on a class of special Lototsky–Bernstein bases. We focus on approximation of | x | on [ − 1 , 1 ] by an approximation process generated from fixed points on Lototsky–Bernstein bases. Our first result shows that the approximation procedure p n (x) to | x | preserves good shapes on [ − 1 , 1 ] . Moreover, some convergence results and inequalities are derived. Our second main result states that the rate convergence of the approximation is O (n − 2) . [ABSTRACT FROM AUTHOR]

In the present paper, we construct a new class of positive linear λ-Bernstein operators based on (p, q)-integers. We obtain a Korovkin type approximation theorem, study the rate of convergence of these operators by using the conception of K-functional and moduli of continuity, and also give a convergence theorem for the Lipschitz continuous functions. [ABSTRACT FROM AUTHOR]

In this paper, we construct a bivariate tensor product generalization of Kantorovich-type Bernstein-Stancu-Schurer operators based on the concept of $(p, q)$ -integers. We obtain moments and central moments of these operators, give the rate of convergence by using the complete modulus of continuity for the bivariate case and estimate a convergence theorem for the Lipschitz continuous functions. We also give some graphs and numerical examples to illustrate the convergence properties of these operators to certain functions. [ABSTRACT FROM AUTHOR]

In this note, we give an elaboration of a basic problem on convergence theorem of $(p, q)$ -analogue of Bernstein-type operators. By some classical analysis techniques, we derive an exact class of $(p_{n},q_{n})$ -integer satisfying $\lim _{n\to\infty }[n]_{p_{n},q_{n}}=\infty$ with $\lim _{n\to\infty}p_{n}=1$ and $\lim _{n\to\infty}q_{n}=1$ under $0< q_{n}< p_{n}\leq1$ . Our results provide an erratum to corresponding results on $(p,q)$ -analogue of Bernstein-type operators that appeared in recent literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called (α,q)-Bernstein operators, denoted by Tn,q,α(f). We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of Tn,q,α(f) to any continuous functions f. The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to f(x). We also obtain the monotonicity with n and q of Tn,q,α(f). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Bézier variant of Kantorovich type λ-Bernstein operators with parameter λ∈[−1,1]. 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. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type λ-Bernstein operators with parameter 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 Bn,λ(f;x) to f(x), and we see that in some cases the errors are smaller than Bn(f) to f. [ABSTRACT FROM AUTHOR]