Abstract
In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’sK-functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-type and associated GBS operators to certain functions with some graphical illustrations and error estimation tables.
Highlights
In [1], Bernstein suggested his polynomials that still inspire many studies today as follows: r Brðμ ; xÞ = 〠 j=0 r j ! x j ð1 − xÞr−jμ j r, x ∈1⁄20, 1, ð1Þ for any r ∈ N and any μ ∈ C1⁄20, 1
Izgi [4] presented a new type of the Bernstein polynomials and studied several approximation results of the univariate and bivariate of these operators
Kajla and Acar [6] constructed a new kind of the α − Bernstein operator and studied a uniform convergence estimate, some direct results involving the asymptotic theorems for these operators
Summary
In [1], Bernstein suggested his polynomials that still inspire many studies today as follows: r. Izgi [4] presented a new type of the Bernstein polynomials and studied several approximation results of the univariate and bivariate of these operators. Kajla and Acar [6] constructed a new kind of the α − Bernstein operator and studied a uniform convergence estimate, some direct results involving the asymptotic theorems for these operators. Acar et al [7] introduced the Kantorovich modifications of the ðp, qÞ − Bernstein operators for bivariate functions using a new ðp, qÞ − integral and obtained the uniform convergence and rate of approximation in terms of modulus of continuity for these operators. Acar and Kajla [9] introduced an extension of the bivariate generalized Bernstein operators with nonnegative real parameters and studied the degree of approximation with regard to Peetre’s K-functional and Lipschitz-type functions. By the help of the Maple software, we give comparisons of the convergence of bivariate of (3) operators and related GBS operators to the certain functions with some graphics and error estimation tables
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