Rates of Ideal Convergence for Approximation Operators

In this paper, we study a general Korovkin-type approximation theory by using the notion of ideal convergence which includes many convergence methods, such as, the usual convergence, statistical convergence, A-statistical convergence, etc. We mainly compute the rate of ideal convergence of sequences of positive linear operators.     

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-Statistical extension of the Korovkin type approximation theorem

In this paper, using the concept ofA-statistical convergence which is a regular (non-matrix) summability method, we obtain a general Korovkin type approximation theorem which concerns the problem of approximating a functionf by means of a sequenceL _n f of positive linear operators.     

I-convergence theorems for a class of k-positive linear operators

In this paper, we obtain some approximation theorems for k- positive linear operators defined on the space of analytical functions on the unit disc, via I-convergence. Some concluding remarks which includes A-statistical convergence are also given.      

Weighted Statistical Relative Invariant Mean in Modular Function Spaces with Related approximation Results

Abstract

The idea of statistical relative convergence on modular spaces has been introduced by Orhan and Demirci. The notion of σ-statistical convergence was introduced by Mursaleen and Edely and further extended based on a fractional order difference operator by Kadak. The concern of this paper is to define two new summability methods for double sequences by combining the concepts of statistical relative convergence and σ-statistical convergence in modular spaces. Furthermore, we give some inclusion relations involving the newly proposed methods and present an illustrative example to show that our methods are nontrivial generalizations of the existing results in the literature. We also prove a Korovkin-type approximation theorem and estimate the rate of convergence by means of the modulus of continuity. Finally, using the bivariate type of Stancu-Schurer-Kantorovich operators, we display an example such that our approximation results are more powerful than the classical, statistical, and relative modular cases of Korovkin-type approximation theorems.     

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.     

On a <Emphasis Type="Italic">q</Emphasis>-analogue of Stancu operators

This paper is concerned with a generalization in q-Calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established.     

Some variations of the Bohman–Korovkin theorem

In this paper we develop the main aspects of the Bohman–Korovkin theorem on approximation of continuous functions with the use of A-statistical convergence and matrix summability method which includes both convergence and almost convergence. Since statistical convergence and almost convergence methods are incompatible we conclude that these methods can be used alternatively to get some approximation results.     

Some approximation results involving the q‐Szász–Mirakjan–Kantorovich type operators via Dunkl's generalization

The purpose of this paper is to introduce a family of q‐Szász–Mirakjan–Kantorovich type positive linear operators that are generated by Dunkl's generalization of the exponential function. We present approximation properties with the help of well‐known Korovkin's theorem and determine the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and the second‐order modulus of continuity. Furthermore, we obtain the approximation results for bivariate q‐Szász–Mirakjan–Kantorovich type operators that are also generated by the aforementioned Dunkl generalization of the exponential function. Copyright © 2017 John Wiley & Sons, Ltd.     

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 Kantorovich-Szász Type Operators Based on Brenke Type Polynomials

In this article, we give a generalization of the Kantorovich-Szász type operators defined by means of the Brenke type polynomials introduced in the literature and obtain convergence properties of these operators by using Korovkin’s theorem. Some graphical examples using the Maple program for this approximation are given. We also establish the order of convergence by using modulus of smoothness and Peetre’s K-functional and give a Voronoskaja type theorem. In addition, we deal with the convergence of these operators in a weighted space.     

A Voronovskaya-type formula for SMK operators via statistical convergence

In this paper, we obtain a statistical Voronovskaya-type theorem for the Szász-Mirakjan-Kantorovich (SMK) operators by using the notion of A-statistical convergence, where A is a non-negative regular summability matrix.     

On Approximation Properties of Generalized q-Bernstein Operators

In this study, we identify a generalization of q-Bernstein type operators and investigate approximation properties of a sequence of these operators . We estimate rate of approximation by modulus of continuity. We prove Voronovskaya type theorem for these operators.     

The generalization of Meyer-König and Zeller operators by generating functions

In this paper, theorems are proved concerned with some approximation properties of generating functions type Meyer-König and Zeller operators with the help of a Korovkin type theorem. Secondly, we compute the rates of convergence of these operators by means of the modulus of continuity, Peetre's K-functional and the elements of the modified Lipschitz class. Also we introduce the rth order generalization of these operators and we obtain approximation properties of them. In the last part, we give some applications to the differential equations.     

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.     

Korovkin-Type Approximation Theorem for Double Sequences of Positive Linear Operators via Statistical A-Summability

In this paper, using the concept of statistical A-summability which is stronger than the A-statistical convergence we provide a Korovkin-type approximation theorem on the space of all continuous real valued functions defined on any compact subset of the real two-dimensional space. We also study the rates of statistical A-summability of positive linear operators.     

A Baskakov type generalization of statistical Korovkin theory

In this paper using the notion of A-statistical convergence, where A is a nonnegative regular summability matrix, we obtain some statistical variants of Baskakov's results on the Korovkin type approximation theorems.     

Statistical Korovkin Theory for Multivariate Stochastic Processes

In this article, we study Korovkin-type approximation theorems for multivariate stochastic processes via the concept of A-statistical convergence. A non-trivial example expressing the importance of our results is also presented.     

Statistical approximation properties of q-Bleimann,Butzer and Hahn operators

The main aim of this study is to introduce a new generalization of q-Bleimann, Butzer and Hahn operators and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. Rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function are also established. Our results show that rates of convergence of our operators are at least as fast as classical BBH operators. The second aim of this study is to construct a bivariate generalization of the operator and also obtain the statistical approximation properties.