GENERALIZED BERNSTEIN-KANTOROVICH OPERATORS OF BLENDING TYPE

In this note, we derive some approximation properties of the generalized Bernstein-Kantorovich type operators based on two nonnegative parameters considered  by A. Kajla [Appl. Math. Comput. 2018]. We establish Voronovskaja type asymptotic theorem for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also derived. Finally, we show the convergence of the operators by illustrative graphics in Mathematica software to certain functions

BLENDING TYPE APPROXIMATION BY BEZIER-SUMMATION-INTEGRAL TYPE OPERATORS

Acar, Tuncer/0000-0003-0982-9459WOS: 000439232800019In this note we construct the Bezier variant of summation integral type operators based on a non-negative real parameter. We present a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation. In the last section, we study the quantitative Voronovskaja type theorem.Research Project of Kirikkale University, BAP (Turkey) [2017/014]The first author is partially supported by Research Project of Kirikkale University, BAP, 2017/014 (Turkey)

Blending type approximation by bivariate generalized Bernstein type operators

In this article we establish an extension of the bivariate generalization of the Bernstein type operators involving parameters. For these operators we obtain a Voronovskaja type and Grüss Voronovskaja type theorems and the degree of approximation by means of the Lipschitz type space. Further, we present the associated Generalized Boolean Sum (GBS) operators and establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MAPLE software. Mathematics Subject Classification (2010): 41A25, 26A15

Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution

In the paper the authors introduce the Kantorovich variant of Stancu operators based on Pólya-Eggenberger distribution. By making use of this new operator, we obtain some indispensable auxiliary results. We also deal with a Voronovskaja type asymptotic formula and some estimates of the rate of approximation involving modulus of smoothness, such as Ditzian-Totik modulus of smoothness. The rate of convergence for differential functions whose derivatives are bounded is also obtained

Modified Bernstein–Durrmeyer Type Operators

We constructed a summation–integral type operator based on the latest research in the linear positive operators area. We establish some approximation properties for this new operator. We highlight the qualitative part of the presented operator; we studied uniform convergence, a Voronovskaja-type theorem, and a Grüss–Voronovskaja type result. Our subsequent study focuses on a direct approximation theorem using the Ditzian–Totik modulus of smoothness and the order of approximation for functions belonging to the Lipschitz-type space. For a complete image on the quantitative estimations, we included the convergence rate for differential functions, whose derivatives were of bounded variations. In the last section of the article, we present two graphs illustrating the operator convergence

Blending type approximation by Stancu-Kantorovich operators based on Polya-Eggenberger distribution

In the paper the authors introduce the Kantorovich variant of Stancu operators based on Polya-Eggenberger distribution. By making use of this new operator, we obtain some indispensable auxiliary results. We also deal with a Voronovskaja type asymptotic formula and some estimates of the rate of approximation involving modulus of smoothness, such as Ditzian-Totik modulus of smoothness. The rate of convergence for differential functions whose derivatives are bounded is also obtained

Approximation by Bivariate Bernstein-Durrmeyer Operators on a Triangle

In the present paper, we obtain some approximation properties for the bivariate Bernstein-Durrmeyer operators on a triangle. We characterize the rate of convergence in terms of K-functional and the usual and second order modulus of continuity. We estimate the order of approximation by Voronovskaja type result and illustrate the convergence of these operators to a certain function through graphics using Mathematica algorithm. We also discuss the comparison of the convergence of the bivariate Bernstein-Durrmeyer operators and the bivariate Bernstein-Kantorovich operators to the function through illustrations using Mathematica. Lastly, we study the simultaneous approximation for first order partial derivatives and the shape preserving properties of these operators

Bézier-Summation-Integral-Type Operators That Include Pólya–Eggenberger Distribution

We define the summation-integral-type operators involving the ideas of Pólya–Eggenberger distribution and Bézier basis functions, and study some of their basic approximation properties. In addition, by means of the Ditzian–Totik modulus of smoothness, we study a direct theorem as well as a quantitative Voronovskaja-type theorem for our newly constructed operators. Moreover, we investigate the approximation of functions with derivatives of bounded variation (DBV) of the aforesaid operators

A Kantorovich variant of a generalized Bernstein operators

In this note we present a Kantorovich variant of the operators proposed by [X. Y. Chen, J. Q. Tan, Z. Liu, J. Xie, J. Math. Anal. Appl., 450 (2017), 244-261] based on non-negative parameters. Here, we prove an approximation theorem with the help of Bohman-Korovkin's principle and study the estimate of the rate of approximation by using the modulus of smoothness and Lipschitz type function for these operators. Also, we establish Voronovskaja type theorem and Korovkin type A-statistical approximation theorem of these operators