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.     

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.     

q‐Voronovskaya type theorems for q‐Baskakov operators

In the present paper, we prove quantitative q‐Voronovskaya type theorems for q‐Baskakov operators in terms of weighted modulus of continuity. We also present a new form of Voronovskaya theorem, that is, q‐Grüss‐Voronovskaya type theorem for q‐Baskakov operators in quantitative mean. Hence, we describe the rate of convergence and upper bound for the error of approximation, simultaneously. Our results are valid for the subspace of continuous functions although classical ones is valid for differentiable functions. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

q-Baskakov型算子的A-统计逼近

引入一类q-Baskakov型算子,对一个非负正则可求和矩阵A,应用A-统计逼近的理论,研究了这类修正的Korovkin型统计逼近性质.对于0q≤1,借助连续性模,证得这类q-Baskakov型算子的收敛速度要优于q-Baskakov算子.     

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.     

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.     

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.     

Asymptotic formulas for generalized Baskakov‐Durrmeyer operators

The paper deals with general Baskakov‐Durrmeyer operators containing several previous definitions as special cases. We construct a new sequence of BaskakovDurrmeyer operators depending on a parameter γ. We present a quantitative Voronovskaya type theorem in terms of weighted modulus of smoothness using sixth‐order central moment. In addition, we studied Grü ss‐type Voronovskaya theorem. All results in this work show that our new operators are flexible and sensitive to the rate of convergence to f, depending on our selection of γ(x).     

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.     

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].     

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.     

Generalized Bernstein‐Kantorovich–type operators on a triangle

In this note, we construct generalized Bernstein‐Kantorovich–type operators on a triangle. The concern of this note is to present a Voronovskaja‐type and Grüss Voronovskaja‐type asymptotic theorems, and some estimates of the rate of approximation with the help of K‐functional, first and second order modulus of continuity. We also obtain Korovkin‐ and Voronovskaja‐type statistical approximation theorems via weighted mean matrix method. Lastly, we show that the numerical results which explain the validity of the theoretical results and the effectiveness of the constructed operators.     

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.     

q‐Bernstein polynomials related to q‐Frobenius–Euler polynomials,l‐functions,and q‐Stirling numbers

The aim of this paper was to derive new identities and relations associated with the q‐Bernstein polynomials, q‐Frobenius–Euler polynomials, l‐functions, and q‐Stirling numbers of the second kind. We also give some applications related to theses polynomials and numbers. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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.     

The convergence of q ‐Bernstein polynomials (0 < q < 1) in the complex plane

The paper focuses at the estimates for the rate of convergence of the q ‐Bernstein polynomials (0 < q < 1) in the complex plane. In particular, a generalization of previously known results on the possibility of analytic continuation of the limit function and an elaboration of the theorem by Wang and Meng is presented (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.