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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 α and λ 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 α and λ , and they propose better approximation results. [ABSTRACT FROM AUTHOR]

In this study, we introduce a Durrmeyer type of Bleimann, Butzer, and Hahn operators (BBH) on (p,q)-integers. We derive the some approximation properties for these operators. We also give some graphs and numerical examples to illustrate the convergence properties of these operators to certain 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 a new kind of Kantorovich-type Bernstein–Stancu–Schurer operators based on the concept of ( p, q )-integers. We investigate statistical approximation properties and establish a local approximation theorem, we also give a convergence theorem for the Lipschitz continuous functions. Finally, we give some graphics and numerical examples to illustrate the convergence properties of operators to some functions. [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]