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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]

The topic of approximation with positive linear operators in contemporary functional analysis and theory of functions has emerged in the last century. One of these operators is Meyer–König and Zeller operators and in this study a generalization of Meyer–König and Zeller type operators based on a function τ by using two sequences of functions will be presented. The most significant point is that the newly introduced operator preserves { 1 , τ , τ 2 } instead of classical Korovkin test functions. Then asymptotic type formula, quantitative results, and local approximation properties of the introduced operators are given. Finally a numerical example performed by MATLAB is given to visualize the provided theoretical results. [ABSTRACT FROM AUTHOR]

In the present paper, we give an operator with the help of a generalization of Boas–Buck type polynomials by means of Gamma function. We have approximation properties and moments. The rate of convergence is given by the Ditzian–Totik first order modulus of smoothness and the K-functional. Furthermore, we obtain the point-wise estimations for this operator. [ABSTRACT FROM AUTHOR]

In the present paper, Kantorovich type λ -Bernstein operators via (p, q)-calculus are constructed, and the first and second moments and central moments of these operators are estimated in order to achieve our main results. An A-statistical convergence theorem and the rate of A-statistical convergence theorems are obtained according to some analysis methods and the definitions of A-statistical convergence, the rate of A-statistical convergence and modulus of smoothness. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a family of bivariate α , q -Bernstein–Kantorovich operators and a family of G B S (Generalized Boolean Sum) operators of bivariate α , q -Bernstein–Kantorovich type. For the former, we obtain the estimate of moments and central moments, investigate the degree of approximation for these bivariate operators in terms of the partial moduli of continuity and Peetre's K-functional. For the latter, we estimate the rate of convergence of these G B S operators for B-continuous and B-differentiable functions by using the mixed modulus of smoothness. [ABSTRACT FROM AUTHOR]