A Korovkin-type theorem for double sequences of positive linear operators via power series method

In this paper, using power series method we obtain a Korovkin type theorem for double sequences of real valued functions defined on a compact subset of \(\mathbb {R}^{2}\)(the real two-dimensional space). We also present an example that satisfies our theorem. Finally, we calculate the rate of convergence.     

Korovkin type approximation theorem for BBH type operators via <Emphasis Type="Italic">J</Emphasis>-convergence

This paper provides a Korovkin type approximation theorem for a class of positive linear operators including Bleimann-Butzer and Hahn operators via J-convergence.     

A Korovkin type approximation theorems via $$\mathcal{I}$$ -convergence

Using the concept of -convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.     

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.      

KOROVKIN TYPE APPROXIMATION THEOREMS ON THE SPACE OF CONTINUOUSLY DIFFERENTIABLE FNCTIONS

It is studied Korovkin type approximation theorems on C⁽¹⁾ ([0, 1]) the space of continuously differentiable functions on the unit interval. It is proved that test functions for which Korovkin type approximation theorems hold depending on norms of C⁽¹⁾ ([0, 1]).     

Matrix Summability and Positive Linear Operators

In the present paper we prove a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space C_ρ1 into a weighted space B_ρ2 with the use of a matrix summability method which includes both convergence and almost convergence. We also study the rates of convergence of these operators.     

Korovkin type approximation theorems on the space of continuously differentiable fnctions

It is studied Korovkin type approximation theorems on C⁽¹⁾ ([0, 1]) the space of continuously differentiable functions on the unit interval. It is proved that test functions for which Korovkin type approximation theorems hold depending on norms of C⁽¹⁾ ([0, 1]).     

Dunkl generalization of Szász beta‐type operators

The goal in the paper is to advertise Dunkl extension of Szász beta‐type operators. We initiate approximation features via acknowledged Korovkin and weighted Korovkin theorem and obtain the convergence rate from the point of modulus of continuity, second‐order modulus of continuity, the Lipschitz class functions, Peetre's K‐functional, and modulus of weighted continuity by Dunkl generalization of Szász beta‐type operators.     

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 Korovkin-Type Approximation Theorem and Power Series Method

In this paper, using power series methods we give an abstract Korovkin type approximation theorem for a sequence of positive linear operators mapping \({C\left(X, \mathbb{R}\right)}\) into itself.     

$$K_{a}$$-convergence and Korovkin type approximation

In the present paper, we study a Korovkin type approximation theorem in the setting of \(K_{a}\)-convergence that contains the classical result. We also study the rate of \(K_{a}\)-convergence and afterwards, we give some concluding remarks.     

Representation,interpolation, and reiteration theorems for generalized approximation spaces

We show that the representation theorem for classical approximation spaces can be generalized to spaces A(X,l ^q (ℬ))={f∈X:{E _n (f)}∈l ^q (ℬ)} in which the weighted l ^q -space l ^q (ℬ) can be (more or less) arbitrary. We use this theorem to show that generalized approximation spaces can be viewed as real interpolation spaces (defined with K-functionals or main-part K-functionals) between couples of quasi-normed spaces which satisfy certain Jackson and Bernstein-type inequalities. Especially, interpolation between an approximation space and the underlying quasi-normed space leads again to an approximation space. Together with a general reiteration theorem, which we also prove in the present paper, we obtain formulas for interpolation of two generalized approximation spaces. Received: December 6, 2001; in final form: April 2, 2002?Published online: March 14, 2003     

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 Korovkin Type Theorem in the Space of Locally Integrable Functions

It is shown that a Korovkin type theorem for a sequence of linear positive operators acting in weighted space L _p,w (loc) does not hold in all this space and is satisfied only on some subspace.     

Szász-Durrmeyer Operators Based on Dunkl Analogue

In this article, we construct Sz\(\acute{a}\)sz-Durrmeyer type operators based on Dunkl analogue. We investigate several approximation results by these positive linear sequences, e.g. rate of convergence by means of classical modulus of continuity, uniform approximation using Korovkin type theorem on compact interval. Further, we discuss local approximations in terms of second order modulus of continuity, Peetre’s K-functional, Lipschitz type class and rth order Lipschitz-type maximal function. Weighted approximation and statistical approximation results are discussed in the last of this article.     

On statistical approximation of a general class of positive linear operators extended in q-calculus

In this paper we present a general class of positive linear operators of discrete type based on q-calculus and we investigate their weighted statistical approximation properties by using a Bohman–Korovkin type theorem. We also mark out two particular cases of this general class of operators.     

Ishikawa iterative process with errors for nonlinear equations of generalized monotone type in Banach spaces

The concept of the operators of generalized monotone type is introduced and iterative approximation methods for a fixed point of such operators by the Ishikawa and Mann iteration schemes {xn} and {yn} with errors is studied. Let X be a real Banach space and T : D ? X → 2^D be a multi‐valued operator of generalized monotone type with fixed points. A new general lemma on the convergence of real sequences is proved and used to show that {xn} converges strongly to a unique fixed point of T in D. This result is applied to the iterative approximation method for solutions of nonlinear equations with generalized strongly accretive operators. Our results generalize many of know results. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rate of convergence in L_{p} approximation

In the present paper we give a Korovkin type approximation theorem for a sequence of positive linear operators acting from \(L_{p}\left[ a,b\right] \) into itself using the concept of \(\mathcal {A}\) -summation processes. We also study the rate of convergence of these operators.     

Approximation in Banach space by linear positive operators

In this paper, we obtain a sufficient condition for the convergence of positive linear operators in Banach function spaces on \({\mathbb {R}}^n\) and derive a Korovkin type theorem for these spaces. Also, we generalized this result via statistical sense.     

Korovkin type approximation theorem via Ka−convergence on weighted spaces

In this paper, we study the Korovkin type approximation theorem for K_a‐ convergence, which is an interesting convergence method on weighted spaces. We also study the rate of K_a?convergence by using the weighted modulus of continuity and afterwards, we present a nontrivial application.