Korovkin Type Research Articles

A Stancu type extension of the Cheney-Sharma Chlodovsky operators

In this paper we introduce a Stancu type extension of the Cheney-Sharma Chlodovsky operators based on the ideas presented by Cătinaș and Buda, Bostanci and Bașcanbaz-Tunca, respectively Söylemez and Tașdelen. For this new operators we study some approximation and convexity properties and the preservation of the Lipschitz constant and order. Finally, we study approximation properties of the new operators with the help of Korovkin type theorems.

Approximation theorems via Pp -statistical convergence on weighted spaces

Abstract In this paper, we obtain some Korovkin type approximation theorems for double sequences of positive linear operators on two-dimensional weighted spaces via statistical type convergence method with respect to power series method. Additionally, we calculate the rate of convergence. As an application, we provide an approximation using the generalization of Gadjiev-Ibragimov operators for Pp -statistical convergence. Our results are meaningful and stronger than those previously given for two-dimensional weighted spaces.

Approximation by Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators

In this research paper, we construct a new sequence of Riemann–Liouville type fractional ‐Bernstein–Kantorovich operators. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first order modulus of continuity of these operators. Further, we study Voronovskaja type theorem, quantitative Voronovskaya type theorem, Chebyshev–Grüss inequality and Grüss–Voronovskaya type theorem.

Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials

<p style='text-indent:20px;'>Karaisa [<xref ref-type="bibr" rid="b29">29</xref>] presented Jakimovski- Leviatan- Durrmeyer type operators by means of Appell polynomials. In a similar manner, Wani et al. [<xref ref-type="bibr" rid="b43">43</xref>] proposed a sequence of Jakimovski-Leviatan-Durrmeyer type operators involving Brenke type polynomials which include Appell polynomials and Hermite polynomials. We note that the definitions of the operators given in both these papers are not correct. In the present article, we introduce a Stancu variant of the operators considered in [<xref ref-type="bibr" rid="b43">43</xref>] after correcting their definition. The definition of the operator proposed in [<xref ref-type="bibr" rid="b29">29</xref>] may be similarly corrected. We establish the Korovkin type approximation theorem and the rate of convergence by means of the usual modulus of continuity, Peetre's K-functional and the class of Lipschitz type functions for our operators. Next, we discuss the Voronovskaja and Gr<inline-formula><tex-math id="M1">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type asymptotic theorems. Finally, we study the convergence of these operators in a weighted space and the Korovkin type weighted statistical approximation theorem.</p>

Fuzzy Korovkin type Theorems via deferred Cesaro and deferred Euler equi-statistical convergence

We establish a fuzzy Korovkin type approximation theorem by using \(eq-stat^{D}_{CE}\)(deferred Ces\'{a}ro and deferred Euler equi-statistical) convergence proposed by Saini et al. for continuous functions over \([a,b]\). Further, we determine the rate of convergence via fuzzy modulus of continuity.

A new type of Szász–Mirakjan operators based on q-integers

In this article, by using the notion of quantum calculus, we define a new type Szász–Mirakjan operators based on the q-integers. We derive a recurrence formula and calculate the moments Phi _{n,q}(t^{m};x) for m=0,1,2 and the central moments Phi _{n,q}((t-x)^{m};x) for m=1,2. We give estimation for the first and second-order central moments. We present a Korovkin type approximation theorem and give a local approximation theorem by using modulus of continuity. We obtain a local direct estimate for the new Szász–Mirakjan operators in terms of Lipschitz-type maximal function of order α. Finally, we prove a Korovkin type weighted approximation theorem.

Theorems of Second Korovkin Type with respect to Triangular $A$-Statistical Convergence

This article is a continuation of our previous works. We mainly investigate a Korovkin type theorem for double sequences of positive linear operators defined in the space of all $2\pi $-periodic and real valued continuous functions on the real two-dimensional space with help of the concept of triangular $A$-statistical convergence, which is a kind of statistical convergence for double real sequences. Also, we analyze the rate of convergence of double operators in this via modulus of continuity.

Some approximation results on Chlodowsky type q−Bernstein-Schurer operators

The main concern of this article is to obtain several approximation features of the new Chlodowsky type q-Bernstein-Schurer operators. We prove the Korovkin type approximation theorem and discuss the order of convergence with regard to the ordinary modulus of continuity, an element of Lipschitz type and Peetre?s K-functional, respectively. In addition, we derive the Voronovskaya type asymptotic theorem. Finally, using of Maple software, we present the comparison of the convergence of Chlodowsky type q-Bernstein-Schurer operators to the certain functions with some graphical illustrations and error estimation tables.

Asymptotically deferred statistical equivalent functions of order $ \alpha $ in amenable semigroups

The aim of this paper is to study the asymptotic convergence of the sequences. Concretely, we will give the asymptotically statistical convergence of sequences. The asymptotically statistical convergence is studied in many papers including papers [3], [5], [23], [26], [29], [39], [41]. In this paper, we introduce and investigate the concepts of asymptotically deferred statistical equivalent functions of order $ \alpha $ and strong asymptotically deferred statistical equivalent functions of order $ \alpha $ defined on discrete countable amenable semigroups, which generalizes some of the results known in the literature. This is achieved by introducing the Folner sequences and sequences of non-negative integers. Based on this concept we apply it to the approximation theory, and we have proved the Korovkin type theorem and study the rate of convergence for positive linear operators which are deferred statistically convergent of order $ \alpha $.

Generalized Ismail-Durrmeyer type operators involving Sheffer polynomials

In this paper, we introduce a new kind of Durrmeyer variant of generalized Ismail operators based on Sheffer polynomials. These type of polynomials are a generalization of Appell polynomials and a special case of Boas-Buck-type polynomials. We study the convergence of these operators with the help of universal Korovkin type theorem, modulus of continuity, Peetre's $ K $-functional and the class of Lipschitz type functions. Furthermore, We extend this study to include Voronovskaja-type asymptotic results, quantitative-Voronovskaja and Gr$ \ddot{u} $ss-Voronovskaja type theorems and the convergence rate of these operators for functions in a polynomial weighted space by means of the weighted modulus of continuity.

Some Korovkin type approximation applications of power series methods

Some Korovkin type approximation applications of power series methods

Some fuzzy Korovkin type approximation theorems via power series summability method

This article provides a power series summability-based Korovkin type approximation theorem for any sequence of fuzzy positive linear operators. Using the notion of fuzzy modulus of smoothness, we also derive an associated approximation theorem concerning the fuzzy rate of convergence of these operators. Furthermore, through an example, we illustrate that our summability- based Korovin type theorem has an advantage over the fuzzy Korovkin type theorem proved in the seminal paper by Anastassiou (Stud. Univ. Babeş-Bolyai Math L(4):3–10, 2005)

Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$

In this paper, we study several approximation properties of Szasz-Mirakjan-Durrmeyer operators with shape parameter λ∈[−1,1]λ∈[−1,1]. Firstly, we obtain some preliminaries results such as moments and central moments. Next, we estimate the order of convergence in terms of the usual modulus of continuity, for the functions belong to Lipschitz type class and Peetre's K-functional, respectively. Also, we prove a Korovkin type approximation theorem on weighted spaces and derive a Voronovskaya type asymptotic theorem for these operators. Finally, we give the comparison of the convergence of these newly defined operators to the certain functions with some graphics and error of approximation table.

AJ-Statistical Approximation of Continuous Functions by Sequence of Convolution Operators

In this paper, following the concept of AI-statistical convergence for real sequences introduced by Savas et al. [22], we deal with Korovkin type approximation theory for a sequence of positive convolution operators defined on C[a, b], the space of all real valued continuous functions on [a, b], in the line of Duman [6]. In the Section 3, we study the rate of AI-statistical convergence.

Statistical weighted [formula omitted]-summability with application to Korovkin's type approximation theorem

In this paper, we first introduce the notion of statistical weighted (Nλ,p,q)(Eλ,1)A-summability and then investigate its relation with the weighted (Nλ,p,q)(Eλ,1)A-statistical convergence. We also study the weighted (Nλ,p,q)(Eλ,1)-regular matrix and find the conditions for the matrix A to be weighted (Nλ,p,q)(Eλ,1)-regular. Furthermore, as an application of the results presented in this paper, we prove a Korovkin type approximation theorem involving the statistical weighted (Nλ,p,q)(Eλ,1)A-summability, which we have introduced in this paper. Relevant connections of the proposed investigation with other recent developments on the subject are also indicated.

On Some Summability Methods for a q-Analogue of an Integral Type Operator Based on Multivariate q-Lagrange Polynomials

The present paper considers a q-analogue of an operator defined by Erkuş-Duman et al. (Calcolo, 45(1):53–67, 2008) involving q-Lagrange polynomials in several variables. The Korovkin type theorems in the settings of deferred weighted A-statistical convergence and the power series method are investigated.

A new weighted statistical convergence and some associated approximation theorems

In this paper we study a notion called (E, 1)(C, 1)−weighted statistical convergence and prove a Korovkin type approximation theorem. The rate of convergence for weighted statistical convergence is obtained. In the last section we also give Voronovskaya and Grüss‐Voronovskaya type theorems and some illustrative numerical examples

A Korovkin Type Approximation Theorem For Balázs Type Bleimann, Butzer and Hahn Operators via Power Series Statistical Convergence

AbstractIn this paper, we obtain a Korovkin type approximation theorem for power series statistical convergence of functions belonging to the class produced by multivariable modulus of continuity function. As an application of this theorem, we construct a non-tensor product Balázs type BBH operator which does not converge in ordinary sense. Moreover, we study promised approximation properties of this operator and compute the rate of convergence. Finally, we prove that our new approximation result works but its classical case fails.

Statistical Convergence via q-Calculus and a Korovkin’s Type Approximation Theorem

In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically C1q-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and statistically C1q-summable sequences. Further, we apply q-statistical convergence to prove a Korovkin type approximation theorem.

Nonlinear versions of Korovkin’s abstract theorems

In this paper we prove Korovkin type theorems for sequences of sublinear, monotone and weak additive operators acting on function spaces C(X), where X is a compact or a locally compact metric space. Our results are illustrated by a series of examples.