In this paper, we introduce a new type λ-Bernstein operators with parameter $\lambda\in[-1,1]$ , 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 $B_{n,\lambda }(f;x)$ to $f(x)$ , and we see that in some cases the errors are smaller than $B_{n}(f)$ to f.

In this paper, we introduce the Bézier variant of Kantorovich type λ-Bernstein operators with parameter $\lambda\in[-1,1]$ . 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.

This paper deals with several approximation properties for a new class of q-Bernstein polynomials based on new Bernstein basis functions with shape parameter $\lambda$ 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.

This paper is devoted to studying the statistical approximation properties of a sequence of univariate and bivariate blending-type Bernstein operators that includes shape parameters $\alpha$ and $\lambda$ and a positive integer. An estimate of the corresponding rates was obtained, and a Voronovskaja-type theorem is given by a weighted A-statistical convergence. A Korovkin-type theorem is provided for the univariate and bivariate cases of the blending-type operators. Moreover, the convergence behavior of the univariate and bivariate new blending basis and new blending operators are exhaustively demonstrated by computer graphics. The studied univariate and bivariate blending-type operators reduce to the well-known Bernstein operators in the literature for the special cases of shape parameters $\alpha$ and $\lambda ,$ and they propose better approximation results.

In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called $(\alpha,q)$ -Bernstein operators, denoted by $T_{n,q,\alpha}(f)$ . We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of $T_{n,q,\alpha}(f)$ 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 $T_{n,q,\alpha}(f)$ .