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.

In this paper, we introduce a new type λ-Bernstein operators with parameter , 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 to , and we see that in some cases the errors are smaller than to f.

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In this paper, we introduce the Bézier variant of Kantorovich type λ-Bernstein operators with parameter . 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 introduce a kind of q-gamma operators based on the concept of a q-integral. We estimate moments of these operators and establish direct and local approximation theorems of the operators. The estimates on the rate of convergence and weighted approximation of the operators are obtained, a Voronovskaya asymptotic formula is also presented. MSC: 41A10; 41A25; 41A36

In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called -Bernstein operators, denoted by . We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of 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 . We also obtain the monotonicity with n and q of .

This paper deals with several approximation properties for a new class of q-Bernstein polynomials based on new Bernstein basis functions with shape parameter λ on the symmetric interval [−1,1]. Firstly, we computed some moments and central moments. Then, we constructed a Korovkin-type convergence theorem, bounding the error in terms of the ordinary modulus of smoothness, providing estimates for Lipschitz-type functions. Finally, with the aid of Maple software, we present the comparison of the convergence of these newly constructed polynomials to the certain functions with some graphical illustrations and error estimation tables.

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. MSC: 41A10; 41A25; 41A36