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.     

Approximation properties of a generalization of positive linear operators

In the present paper we introduce a generalization of positive linear operators and obtain its Korovkin type approximation properties. The rates of convergence of this generalization is also obtained by means of modulus of continuity and Lipschitz type maximal functions. The second purpose of this paper is to obtain weighted approximation properties for the generalization of positive linear operators defined in this paper. Also we obtain a differential equation so that the second moment of our operators is a particular solution of it. Lastly, some Voronovskaja type asymptotic formulas are obtained for Meyer-König and Zeller type and Bleimann, Butzer and Hahn type operators.     

Equivalent theorems on simultaneous approximation by combinations of Gamma operators

In this paper,some equivalent theorems on simultaneous approximation for combinations of Gamma operators by weighted moduli of smoothness ωr(4)λ(f,t)w(4)s(O≤λ≤1)are given.The relation between derivatives of combinations of Gamma operators and smoothness of derivatives of functions is also investigated.     

Approximation of functions of two variables by certain linear positive operators

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an rth order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and find the rate of this convergence using weighted modulus of continuity.     

Approximation by a class of modified Szász-Durrmeyer operators

We modify Szász-Durrmeyer operators by means of three-diagonal generalized matrix which overcomes a difficulty in extending a Berens-Lorentz result to the Szász-Durrmeyer operators for second order of smoothness. The direct and converse theorems for these operators inL _p are also presented by Ditzian-Totik modulus of smoothness.This project is supported by Zhejiang Provincial Foundation of China.     

Pointwise Estimate for Linear Combinations of Bernstein-Kantorovich Operators

For linear combinations of Bernstein-Kantorovich operators K_n, r(f, x), we give an equivalent theorem with ω^2r_?^λ(f, t). The theorem unites the corresponding results of classical and Ditzian-Totik moduli of smoothness.     

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.     

Computational scheme for invariantly unbiased estimation of linear operators in a given class

The values of linear operators of a given class are estimated in the case of measurements including piecewise continuous noise of deterministic structure with unknown parameters. A computational scheme producing unbiased linear estimates that are invariant under the noise is developed. An illustrative example is presented.     

Baskakov—Durrmeyer算子的同时逼近

本文讨论Baskakov-Durrmeyer算子对具有指数型增长的第一类间断点函数及其导数的逼近。     

一类Bernstein型算子线性组合加Jacobi权的逼近

给出了一类用广义杨辉三角阵定义的Bernstein型算子线性组合加Ja-cob i权的逼近在一致逼近意义下的特征刻划.     

Lacunary equi-statistical convergence of positive linear operators

In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform convergence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical convergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied.      

Approximation theorems for a general class of truncated operators

In this paper we give direct approximation theorems for a general family of truncated operators. We discuss the linear and nonlinear cases.     

Bernstein算子导数与高阶光滑性

借助于r阶光滑模ωφ^r(f,t)φ是一般的步权函数，给出了Bernstein算子导数与函数高阶光滑性之间的等价关系。     

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.     

Approximation of functions by certain nonlinear integral operators

We study approximation properties of certain nonlinear integral operators L _n * obtained by a modification of given operators L _n . The operators L _n;r and L _n;r * of r-times differentiable functions are also studied. We give theorems on approximation orders of functions by these operators in polynomial weight spaces.     

Szász型算子线性组合的同时逼近

本文利用ω^2r_φλ(f,t)代替ω^r_φλ(f,t)给出了Szász-Kantorovich算子线性组合同时逼近的估计。     

A certain class of weighted statistical convergence and associated Korovkin‐type approximation theorems involving trigonometric functions

The subject of statistical convergence has attracted a remarkably large number of researchers due mainly to the fact that it is more general than the well‐established theory of the ordinary (classical) convergence. In the year 2013, Edely et al 17 introduced and studied the notion of weighted statistical convergence. In our present investigation, we make use of the (presumably new) notion of the deferred weighted statistical convergence to present Korovkin‐type approximation theorems associated with the periodic functions , and defined on a Banach space . In particular, we apply our concept of the deferred weighted statistical convergence with a view to proving a Korovkin‐type approximation theorem for periodic functions and also to demonstrate that our result is a nontrivial extension of several known Korovkin‐type approximation theorems which were given in earlier works. Moreover, we establish another result for the rate of the deferred weighted statistical convergence for the same set of functions. Finally, we consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.     

Strong type of Steckin inequality for the linear combination of Bernstein operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein operators, which gives the optimal approximation rate. Moreover, a method to prove lower estimates for linear operators is introduced. As a result the lower estimate for the linear combinations of Bernstein operators is obtained by using the Ditzian–Totik modulus of smoothness.