A new kind of Bi-variate $ \lambda $ -Bernstein-Kantorovich type operator with shifted knots and its associated GBS form

<p style='text-indent:20px;'>In this paper, we introduce a bi-variate case of a new kind of <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich type operator with shifted knots defined by Rahman et al. [<xref ref-type="bibr" rid="b31">31</xref>]. The rate of convergence of the bi-variate operators is obtained in terms of the complete and partial moduli of continuity. Next, we give an error estimate in the approximation of a function in the Lipschitz class and establish a Voronovskaja type theorem. Also, we define the associated GBS(Generalized Boolean Sum) operators and study the degree of approximation of Bögel continuous and Bögel differentiable functions by these operators with the aid of the mixed modulus of smoothness. Finally, we show the rate of convergence of the bi-variate operators and their GBS case for certain functions by illustrative graphics and tables using MATLAB algorithms.</p>

Better numerical approximation by λ-Durrmeyer-Bernstein type operators

The main object of this paper is to construct a new Durrmeyer variant of the ?-Bernstein type operators which have better features than the classical one. Some results concerning the rate of convergence in terms of the first and second moduli of continuity and asymptotic formulas of these operators are given. Moreover, we define a bivariate case of these operators and investigate the approximation degree by means of the total and partial modulus of continuity and the Peetre?s K-functional. A Voronovskaja type asymptotic and Gr?ss-Voronovskaja theorem for the bivariate operators is also proven. Further, we introduce the associated GBS (Generalized Boolean Sum) operators and determine the order of convergence with the aid of the mixed modulus of smoothness for the B?gel continuous and B?gel differentiable functions. Finally the theoretical results are analyzed by numerical examples.

On the Approximation Properties of q − Analogue Bivariate λ -Bernstein Type Operators

In this article, we establish an extension of the bivariate generalization of the q -Bernstein type operators involving parameter λ and extension of GBS (Generalized Boolean Sum) operators of bivariate q -Bernstein type. For the first operators, we state the Volkov-type theorem and we obtain a Voronovskaja type and investigate the degree of approximation by means of the Lipschitz type space. For the GBS type operators, we establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate q -Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MATLAB software.

Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators

We construct the bivariate form of Bernstein–Schurer operators based on parameter α. We establish the Voronovskaja-type theorem and give an estimate of the order of approximation with the help of Peetre’s K-functional of our newly defined operators. Moreover, we define the associated generalized Boolean sum (shortly, GBS) operators and estimate the rate of convergence by means of mixed modulus of smoothness. Finally, the order of approximation for Bögel differentiable function of our GBS operators is presented.

Quantitative estimates for the tensor product (p,q)-Balázs-Szabados operators and associated generalized Boolean sum operators

In this study, we give some approximation results for the tensor product of (p,q)-Bal?zs-Szabados operators associated generalized Boolean sum (GBS) operators. Firstly, we introduce tensor product (p,q)-Bal?zs-Szabados operators and give an uniform convergence theorem of these operators on compact rectangular regions with an illustrative example. Then we estimate the approximation for the tensor product (p,q)-Bal?zs-Szabados operators in terms of the complete modulus of continuity, the partial modulus of continuity, Lipschitz functions and Petree?s K-functional corresponding to the second modulus of continuity. After that, we introduce the GBS operators associated the tensor product (p,q)-Bal?zs-Szabados operators. Finally, we improve the rate of smoothness by the mixed modulus of smoothness and Lipschitz class of B?gel continuous functions for the GBS operators.

Bivariate α,q-Bernstein–Kantorovich Operators and GBS Operators of Bivariate α,q-Bernstein–Kantorovich Type

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.

Bivariate q-Bernstein–Chlodowsky–Durrmeyer type operators and the associated GBS operators

This paper deals with the approximation properties of the [Formula: see text]-bivariate Bernstein–Chlodowsky operators of Durrmeyer type. We investigate the approximation degree of the [Formula: see text]-bivariate operators for continuous functions in Lipschitz space and also with the help of partial modulus of continuity. Further, the Generalized Boolean Sum (GBS) operator of these bivariate [Formula: see text]–Bernstein–Chlodowsky–Durrmeyer operators is introduced and the rate of convergence in the Bögel space of continuous functions by means of the Lipschitz class and the mixed modulus of smoothness is examined. Furthermore, the convergence and its comparisons are shown by illustrative graphics for the [Formula: see text]-bivariate operators and the associated GBS operators to certain functions using Maple algorithms.