Kantorovich variant of a new kind of q‐Bernstein–Schurer operators

Ren and Zeng (2013) introduced a new kind of q‐Bernstein–Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K‐functional. Next, we introduce the bivariate case of q‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K‐functional. Finally, we define the generalized Boolean sum operators of the q‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.     

Some approximation properties by a class of bivariate operators

Starting with the well‐ known Bernstein operators, in the present paper, we give a new generalization of the bivariate type. The approximation properties of this new class of bivariate operators are studied. Also, the extension of the proposed operators, namely, the generalized Boolean sum (GBS) in the Bögel space of continuous functions is given. In order to underline the fact that in this particular case, GBS operator has better order of convergence than the original ones, some numerical examples are provided with the aid of Maple soft. Also, the error of approximation for the modified Bernstein operators and its GBS‐type operator are compared.     

Voronovskaya‐type theorems for Urysohn type nonlinear Bernstein operators

The concern of this paper is to continue the investigation of convergence properties of nonlinear approximation operators, which are defined by Karsli. In details, the paper centers around Urysohn‐type nonlinear counterpart of the Bernstein operators. As a continuation of the study of Karsli, the present paper is devoted to obtain Voronovskaya‐type theorems for the Urysohn‐type nonlinear Bernstein operators.     

Construction of a new family of Bernstein‐Kantorovich operators

In the present paper, we construct a new sequence of Bernstein‐Kantorovich operators depending on a parameter α. The uniform convergence of the operators and rate of convergence in local and global sense in terms of first‐ and second‐order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of Bernstein‐Kantorovich operators and their approximation behaviors.     

Blending type approximation by q‐generalized Boolean sum of Durrmeyer type

This paper is in continuation of the work performed by Kajla et al. (Applied Mathematics and Computation 2016; 275 : 372–385.) wherein the authors introduced a bivariate extension of q‐Bernstein–Schurer–Durrmeyer operators and studied the rate of convergence with the aid of the Lipschitz class function and the modulus of continuity. Here, we estimate the rate of convergence of these operators by means of Peetre's K‐functional. Then, the associated generalized Boolean sum operator of the q‐Bernstein–Schurer–Durrmeyer type is defined and discussed. The smoothness properties of these operators are improved with the help of mixed K‐functional. Furthermore, we show the convergence of the bivariate Durrmeyer‐type operators and the associated generalized Boolean sum operators to certain functions by illustrative graphics using Maple algorithm. Copyright © 2017 John Wiley & Sons, Ltd.     

Approximation properties for King's type modified q‐Bernstein–Kantorovich operators

In the present research article, we introduce the King's type modification of q‐Bernstein–Kantorovich operators and investigate some approximation properties. We show comparisons and present some illustrative graphics for the convergence of these operators to some function. Copyright © 2015 John Wiley & Sons, Ltd.     

On simultaneous approximation of the Bernstein Durrmeyer operators

The present paper deals with the study of the rate of convergence of the Bézier variant of certain Bernstein Durrmeyer type operators in simultaneous approximation.     

Some approximation properties of a Durrmeyer variant of q‐Bernstein–Schurer operators

In this paper, we will propose a Durrmeyer variant of q‐Bernstein–Schurer operators. A Bohman–Korovkin‐type approximation theorem of these operators is considered. The rate of convergence by using the first modulus of smoothness is computed. The statistical approximation of these operators is also studied. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximation properties of λ‐Bernstein‐Kantorovich operators with shifted knots

In the present article, Kantorovich variant of λ‐Bernstein operators with shifted knots are introduced. The advantage of using shifted knot is that one can do approximation on [0,1] as well as on its subinterval. In addition, it adds flexibility to operators for approximation. Some basic results for approximation as well as rate of convergence of the introduced operators are established. The r^th order generalization of the operator is also discussed. Further for comparisons, some graphics and error estimation tables are presented using MATLAB.     

Certain positive linear operators with better approximation properties

The present paper deals with a new positive linear operator which gives a connection between the Bernstein operators and their genuine Bernstein‐Durrmeyer variants. These new operators depend on a certain function τ defined on [0,1] and improve the classical results in some particular cases. Some approximation properties of the new operators in terms of first and second modulus of continuity and the Ditzian‐Totik modulus of smoothness are studied. Quantitative Voronovskaja–type theorems and Grüss‐Voronovskaja–type theorems constitute a great deal of interest of the present work. Some numerical results that compare the rate of convergence of these operators with the classical ones and illustrate the relevance of the theoretical results are given.     

A family of univariate rational Bernstein operators

We define and study a new family of univariate rational Bernstein operators. They are positive operators exact on linear polynomials. Moreover, like classical polynomial Bernstein operators, they enjoy the traditional shape preserving properties and they are total variation diminishing. Finally, for a specific class of denominators, some convergence results are proved, in particular a Voronovskaja theorem, and some error bounds are given.     

Bernstein型算子同时逼近误差

该文证明了C[0,1]空间中的函数及其导数可以用Bernstein算子的线性组合同时逼近,得到逼近的正定理与逆定理.同时,也证明了Bernstein算子导数与函数光滑性之间的一个等价关系.该文所获结果沟通了Bernstein算子同时逼近的整体结果与经典的点态结果之间的关系.     

Higher order limit q‐Bernstein operators

In this paper we give the estimates of the central moments for the limit q‐Bernstein operators. We introduce the higher order generalization of the limit q‐Bernstein operators and using the moment estimations study the approximation properties of these newly defined operators. It is shown that the higher order limit q‐Bernstein operators faster than the q‐Bernstein operators for the smooth functions defined on [0, 1]. Copyright © 2011 John Wiley & Sons, Ltd.     

Quantitative estimations of bivariate summation‐integral–type operators

In this paper, we study the approximation properties of bivariate summation‐integral–type operators with two parameters . The present work deals within the polynomial weight space. The rate of convergence is obtained while the function belonging to the set of all continuous and bounded function defined on ([0],∞)(×[0],∞) and function belonging to the polynomial weight space with two parameters, also convergence properties, are studied. To know the asymptotic behavior of the proposed bivariate operators, we prove the Voronovskaya type theorem and show the graphical representation for the convergence of the bivariate operators, which is illustrated by graphics using Mathematica. Also with the help of Mathematica, we discuss the comparison by means of the convergence of the proposed bivariate summation‐integral–type operators and Szász‐Mirakjan‐Kantorovich operators for function of two variables with two parameters to the function. In the same direction, we compute the absolute numerical error for the bivariate operators by using Mathematica and is illustrated by tables and also the comparison takes place of the proposed bivariate operators with the bivariate Szász‐Mirakjan operators in the sense of absolute error, which is represented by table. At last, we study the simultaneous approximation for the first‐order partial derivative of the function.     

Bernstein 算子的同时逼近

借助于D itzian-T otik光滑模研究了Bernstein算子的同时逼近问题,给出了Bernstein算子同时逼近的正定理和等价定理.     

On an extremal relation of Bernstein operators

In this note we present a new characterization of Bernstein operators by showing that they are the only solution of a certain extremal relation.     

Approximation properties of Szász‐Mirakyan‐Kantorovich type operators

In this paper, we introduce and study new type Szász‐Mirakyan‐Kantorovich operators using a technique different from classical one. This allow to analyze the mentioned operators in terms of exponential test functions instead of the usual polynomial type functions. As a first result, we prove Korovkin type approximation theorems through exponential weighted convergence. The rate of convergence of the operators is obtained for exponential weights.     

多元Bernstein算子的导数与函数的光滑性

利用一个新的光滑性度量刻画多元Bernstein算子方向导数的特征,建立Bernstein算子的导数与逼近函数光滑性之间的等价关系。同时,一个关于一元Bernstein算子的相应结果被推广到多元情形。     

Rate of Convergence in Trotter’s Approximation Theorem

We give a quantitative estimate of the convergence in Trotter’s approximation theorem on the convergence of iterates of linear operators to an assigned semigroup. An application is given concerning the classical Bernstein operator on the d-dimensional simplex.