Type Approximation Theorem Research Articles

Approximation by a new sequence of operators involving Apostol-Genocchi polynomials

Abstract The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discussed the approximation results.

Korovkin type approximation via statisticale-convergence on two dimensional weighted spaces

AbstractIn the present work, we prove a Korovkin theorem for statisticale-convergence on two dimensional weighted spaces. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. We also study the rate of statisticale-convergence by using the weighted modulus of continuity and afterwards we present an application in support of our result.

Korovkin type approximation theorem via lacunary equi-statistical convergence in fuzzy spaces

In the present paper, we establish relations between equi-statistical convergence and lacunary equi-statistical convergence of sequences of fuzzy number valued functions. We make an effort to prove Korovkin type approximation theorem via lacunary equi-statistical convergence in fuzzy spaces. Further, we study rates of lacunary equi-statistical fuzzy convergence by using fuzzy modulus of continuity.

Statistical Voronoi mean and applications to approximation theorems

In this paper, we give statistical Voronoi mean which is a new statistical summability method, is not need to be regular and positive. We prove a Korovkin type approximation theorem via this method that covers many important summability methods scattered in the literature. Also, we demonstrate that our theorem is stronger than proven by earlier authors with an interesting application. Finally, we establish the rate of convergence.

A note on the Choquet type operators

In this note the Choquet type operators are introduced, in connection to Choquet's theory of integrability with respect to a not necessarily additive set function. Based on their properties, a quantitative estimate for the nonlinear Korovkin type approximation theorem associated to Bernstein-Kantorovich-Choquet operators is proved. The paper also includes a large generalization of H\"{o}lder's inequality within the framework of monotone and sublinear operators acting on spaces of continuous functions.

Statistical relative uniform convergence of a double sequence of functions at a point and applications to approximation theory

In the present paper, we introduce a new kind of convergence, called the statistical relative uniform convergence, for a double sequence of functions at a point, where the relative uniform convergence of the set of the neighborhoods of the given point is considered. By the use of the statistical relative uniform convergence, we investigate a Korovkin type approximation theorem which makes the proposed method stronger than the ones studied before. After that, we give an example using this new type of convergence. We also study the rate of convergence of the proposed convergence.

Extended Bernstein-Kantorovich-Stancu Operators with Multiple Parameters and Approximation Properties

In this paper, we introduce a novel extension of the Bernstein-Kantorovich-Stancu type operator of degree n with the help of multiple shape parameters. Voronovskaja and Grüss-Voronovskaja type approximation theorems are examined via Ditzian-Totik moduli of smoothness. We investigate basic statistical convergence properties with respect to a non-negative regular summability matrix. Moreover, using Ditzian-Totik moduli, local and global approximation properties associated to the proposed operator have been established. Finally, several illustrative examples are presented to demonstrate the efficiency, applicability and validity of the operator. The graphical and numerical results verify that the proposed operator gives better approximation as well as expand the previous Bernstein-Kantorovich type modifications including single parameter.

Szász type operators involving Charlier polynomials and approximation properties

Our aim is to define modified Sz?sz type operators involving Charlier polynomials and obtain some approximation properties. We prove some results on the order of convergence by using the modulus of smoothness and Peetre?s K-functional. We also establish Voronoskaja type theorem for these operators. Moreover, we prove a Korovkin type approximation theorem via q-statistical convergence.

On Durrmeyer Typeλ-Bernstein Operators via (p,q)-Calculus

In the present paper, Durrmeyer typeλ-Bernstein operators via (p,q)-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in someλcases than Durrmeyer type (p,q)-Bernstein operators.

Korovkin type approximation on an infinite interval via generalized matrix summability method using ideal

Following the notion of $A^\mathcal{I}$-summability method for real sequences \cite{espdsd2} we establish a Korovkin type approximation theorem for positive linear operators on $UC_{*}[0,\infty)$, the Banach space of all real valued uniform continuous functions on $ [0,\infty)$ with the property that $\displaystyle{\lim_{x\rightarrow \infty}f(x)}$ exists finitely for any $f\in UC_{*}[0,\infty)$. In the last section, we extend the Korovkin type approximation theorem for positive linear operators on $UC_{*}\left([0,\infty)\times[0,\infty)\right)$. We then construct an example which shows that our new result is stronger than its classical version.

A-Summation process in the space of locally integrable functions

In this paper, using the concept of summation process, we give a Korovkin type approximation theorem for a sequence of positive linear operators acting from Lp;q (loc) ; the space of locally integrable functions, into itself. We also study rate of convergence of these operators.

A Fractalization of Rational Trigonometric Functions

In Navascues (Z Anal Anwend 25(2):401–418, 2005) and Viswanathan and Chand (J Approx Theory 185:31–50, 2014), new approximation classes of self-referential functions are introduced as fractal versions of the classes of polynomials and rational functions. As a sequel, in the present article, we define a new approximation class consisting of self-referential functions, referred to as the fractal rational trigonometric functions. We establish Weierstrass type approximation theorems for this class and prove the existence of a best fractal rational trigonometric approximant to a real-valued continuous function on a compact interval. Furthermore, we provide an upper bound for the smallest error in approximating a prescribed continuous function by a fractal rational trigonometric function. This extemporizes an analogous result in the context of fractal rational function appeared in Viswanathan and Chand (J Approx Theory 185:31–50, 2014) and followed in the setting of Bernstein fractal rational functions in Vijender (Fractals 26(4):1850045, 2018). The last part of the article aims to clarify and correct the mathematical errors in some results on the Bernstein $$\alpha $$-fractal functions appeared recently in the literature (Vijender in Acta Appl Math 159:11–27, 2019, Vijender in Fractals 26(4):1850045, 2018, Vijender in Mediterr J Math 211:1–16, 2018).

Convergence of \u03bb-Bernstein operators based on (p, q)-integers

In the present paper, we construct a new class of positive linear λ-Bernstein operators based on (p, q)-integers. We obtain a Korovkin type approximation theorem, study the rate of convergence of these operators by using the conception of K-functional and moduli of continuity, and also give a convergence theorem for the Lipschitz continuous functions.

Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence

In this paper, we introduce the concepts of αβ−statistical convergence and strong αβ− summability of double sequences and investigate the relation between these two new concepts. Moreover, statistical convergence and αβ− statistical convergence of double sequences are compared under some certain assumptions. Finally, as an application, we prove Korovkin type approximation theorem for a function of two variables by using the notion of αβ−statistical convergence.

Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems

In the proposed paper, we have introduced the notion of point-wise relatively statistical convergence, relatively equi-statistical convergence and relatively uniform statistical convergence of sequences of functions based on the difference operator of fractional order including (p, q)-gamma function via the deferred Norlund mean. As an application point of view, we have proved a Korovkin type approximation theorem by using the relatively deferred Norlund equi-statistical convergence of difference sequences of functions and intimated that our theorem is a generalization of some well-established approximation theorems of Korovkin type which was presented in earlier works. Moreover, we estimate the rate of the relatively deferred Norlund equi-statistical convergence involving a non-zero scale function. Furthermore, we use the modulus of continuity to estimate the rate of convergence of approximating positive linear operators. Finally, we set up various fascinating examples in connection with our results and definitions presented in this paper.

Lambda^2-statistical convergence and its applicationto Korovkin second theorem

In this paper, we use the notion of strong $(N, \lambda^2)-$summability to generalize the concept of statistical convergence. We call this new method a $\lambda^2-$statistical convergence and denote by $S_{\lambda^2}$ the set of sequences which are $\lambda^2-$statistically convergent. We find its relation to statistical convergence and strong $(N, \lambda^2)-$summability. We will define a new sequence space and will show that it is Banach space. Also we will prove the second Korovkin type approximation theorem for $\lambda^2$-statistically summability and the rate of $\lambda^2$-statistically summability of a sequence of positive linear operators defined from $C_{2\pi}(\mathbb{R})$ into $C_{2\pi}(\mathbb{R}).$

Relative modular uniform approximation by means of the power series method with applications

We introduce the notion of relative convergence by means of a four dimensional matrix in the sense of the power series method, which includes Abel's as well as Borel's methods, to prove a Korovkin type approximation theorem by using the test functions {1,y,z,y2+z2} and a double sequence of positive linear operators defined on modular spaces. We also endeavor to examine some applications related to this new type of approximation.

On ideal summability and a Korovkin type approximation theorem

In this paper, we define and study the notion of $$I_{\lambda }$$ -convergence as a variant of the notion of ideal convergence, where $$\lambda =(\lambda _{n})$$ is a nondecreasing sequence of positive real numbers such that $$\lambda _{n+1} \le \lambda _{n}+1,\lambda _{1}=1,\lambda _{n}\rightarrow \infty (n\rightarrow \infty ).$$ We further apply this notion of summability to prove a Korovkin type approximation theorem.

Abel Convergence of the Sequence of Positive Linear Operators in $L_{p,q}\left( loc\right) $

In this paper, we study a Korovkin type approximation theorem for a sequence of positive linear operators acting from $L_{p,q}\left( loc\right) $ into itself with the use of Abel method which is a sequence-to-function transformation. Using the modulus of continuity for $L_{p,q}\left( loc\right) $ we also give the rate of Abel convergence of these operators.

Statistical Convergence with Respect to Power Series Methods and Applications to Approximation Theory

In this article we introduce the concepts of -statistical convergence and Pp-strong convergence that are introduced via power series methods. Introducing a new type of uniform integrability with the help of power series method we obtain a relationship between these concepts which is actually a characterization of the concept of Pp-strong convergence. As an application we give a Korovkin type approximation theorem by using Pp-statistical convergence and also present an example to illustrate that Pp-statistical approximation is stronger than uniform approximation.