Statistical approximation properties of the Durrmeyer type q‐Bleimann,Butzer, and Hahn operators

In this paper, we introduce a Durrmeyer‐type generalization of q‐Bleimann, Butzer, and Hahn operators based on q‐integers and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. We also compute rates of statistical convergence of these q‐type operators by means of the modulus of continuity and Lipschitz‐type maximal function, respectively. Copyright © 2011 John Wiley & Sons, Ltd.     

Statistical approximation properties of q-Baskakov-Kantorovich operators

In the present paper we introduce a q-analogue of the Baskakov-Kantorovich operators and investigate their weighted statistical approximation properties. By using a weighted modulus of smoothness, we give some direct estimations for error in case 0 < q < 1.     

Korovkin type theorem for non‐tensor Balázs type Bleimann,Butzer and Hahn operators

In this paper, we prove a certain Korovkin type approximation theorem by introducing new test functions. We introduce the non‐tensor Balázs type Bleimann, Butzer and Hahn operators and give the approximation property by using this new Korovkin theorem. Furthermore, we obtain the rate of convergence of these operators by means of modulus of continuity. Finally, we state the multivariate version of the abovementioned Korovkin type theorem. Copyright © 2014 John Wiley & Sons, Ltd.     

A note on a Bernstein-type operator of Bleimann, Butzer, and Hahn

Let L_n(f, x)(f ε C)[0, ∞)) be a Bernstein-type approximation operator as defined and studied by Bleimann, Butzer, and Hahn. Probabilistic arguments are used to simplify and sharpen some of their results. The rates of convergence are given in terms of the first and second moduli of continuity. Moreover, an appropriate limit of L_n is shown to be the well-known Szasz operator.     

Rate of convergence for the Bézier variant of the Bleimann–Butzer–Hahn operators

In the present paper, the authors make use of some results in probability theory with a view to estimating the rate of convergence for the Bézier variant of the Bleimann–Butzer–Hahn operators for functions of bounded variation.     

一类修正的Durrmeyer型q-Baskakov算子的统计收敛性质

本文基于q-积分的概念,定义了一类新的修正的Durrmeyer型q-Baskakov算子,应该指出该算子不同于Aral和Gupta(2010)所定义的算子.通过计算得到算子的各阶矩量及中心矩,研究了算子的统计收敛性质并得到Voronovskaya型渐近展开公式.     

Statistical approximation properties of high order operators constructed with the Chan-Chyan-Srivastava polynomials

In this paper, by including high order derivatives of functions being approximated, we introduce a general family of the linear positive operators constructed by means of the Chan-Chyan-Srivastava multivariable polynomials and study a Korovkin-type approximation result with the help of the concept of A-statistical convergence, where A is any non-negative regular summability matrix. We obtain a statistical approximation result for our operators, which is more applicable than the classical case. Furthermore, we study the A-statistical rates of our approximation via the classical modulus of continuity.     

A Best Asymptotic Constant Related to the Operators of Bleimann, Butzer and Hahn

We determine the best asymptotic constant in an estimation of the second central moment of the Bleimann, Butzer and Hahn operators L _n yielding some new results concerning their rate of convergence. In particular, we find $$\lim_{n\in\mathbb{N}}\sup_{0 < x < \infty}\frac{n\,L_{n}((\cdot-x)^{2} ;x)}{x(1+x)^{2}}$$ .     

Statistical approximation theorems by k-positive linear operators

In this paper, using A-statistical convergence we obtain various approximation theorems by means of k-positive linear operators defined on the space of all analytic functions on the unit disk. Received: 17 February 2005     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

Statistical approximation by positive linear operators on modular spaces

In this paper, we investigate the problem of statistical approximation to a function by means of positive linear operators defined on a modular space. Especially, in order to get more powerful results than the classical aspects we mainly use the concept of statistical convergence. A non-trivial application is also presented.     

Modified Kantorovich operators with better approximation properties

Numerical Algorithms - In the present paper, we study a new kind of Bernstein-Kantorovich-type operators. Here, we discuss a uniform convergence estimate for this modified form. Also, some direct...     

Averaging of linear operators, adiabatic approximation, and pseudodifferential operators

An example of Schr?dinger and Klein-Gordon equations with fast oscillating coefficients is used to show that they can be averaged by an adiabatic approximation based on V. P. Maslov??s operator method.     

Some approximation properties of q-Baskakov-Durrmeyer operators

Very recently Aral and Gupta [1] introduced q analogue of Baskakov-Durrmeyer operators. In the present paper we extend the studies, we establish the recurrence relations for the central moments and obtain an asymptotic formula. Also in the end we propose modified q-Baskakov-Durrmeyer operators, from which one can obtain better approximation results over compact interval.     

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.     

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.     

On approximation properties of non-convolution type nonlinear integral operators

In the present paper we state some approximation theorems concerning pointwise convergence and its rate for a class of non-convolution type nonlinear integral operators of the form:Tλ (f;x) = B A Kλ (t,x, f (t))dt , x ∈< a,b >, λ∈Λ. In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 of f as (x,λ ) → (x0,λ0) in L1 < A,B >, where < a,b > and < A,B > are is an arbitrary intervals in R, Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology. The results of the present paper generalize several ones obtained previously in the papers [19]-[23].