Zeller Operators Research Articles

Parametric generalization of the [formula omitted]-Meyer–König–Zeller operators

In this paper, we introduce α parametric generalization of the Meyer–König–Zeller operators based on q-integer, which provides better error estimation then the usual q-MKZ operators. We obtain a differential recurrence relation satisfied by (α,q)-Meyer–König and Zeller operators and by using it, we calculate the moments t1−tm, m∈0,1,2,3,4 more efficiently. We compute the error of approximation by means of the modulus of continuity and modified Lipschitz class functionals. We further derive the Riccati differential equation satisfied by the first three moments. Graphical and numerical illustrative examples are also given to show the power of approximation with these new operators. Finally, an illustrative real-world example associated with the surface air temperature been investigated to demonstrate the modeling capabilities of these operators.

Approximation by nonlinear Meyer-König and Zeller operators based on q-integers

Abstract In this paper, we introduce the nonlinear Meyer-König and Zeller operators based on q-integers. Firstly, we describe the q-Meyer-König and Zeller operators of max-product type. Then, we give an error estimation for the q-Meyer-König and Zeller operators of max-product kind by using a modulus of continuity.

Variational Approximation for Modified Meyer-König and Zeller Operators

Variational Approximation for Modified Meyer-König and Zeller Operators

A Note on New Construction of Meyer-König and Zeller Operators and Its Approximation Properties

The topic of approximation with positive linear operators in contemporary functional analysis and theory of functions has emerged in the last century. One of these operators is Meyer–König and Zeller operators and in this study a generalization of Meyer–König and Zeller type operators based on a function τ by using two sequences of functions will be presented. The most significant point is that the newly introduced operator preserves {1,τ,τ2} instead of classical Korovkin test functions. Then asymptotic type formula, quantitative results, and local approximation properties of the introduced operators are given. Finally a numerical example performed by MATLAB is given to visualize the provided theoretical results.

Rates of Power Series Statistical Convergence of Positive Linear Operators and Power Series Statistical Convergence of $$\boldsymbol{q}$$-Meyer–Köni̇g and Zeller Operators

Rates of Power Series Statistical Convergence of Positive Linear Operators and Power Series Statistical Convergence of $$\boldsymbol{q}$$-Meyer–Köni̇g and Zeller Operators

Approximation by using the Meyer-König and Zeller operators based on (p,q)-analogue

In this paper, a generalization of the q-Meyer-K?nig and Zeller operators by means of the (p,q)- calculus is introduced. Some approximation results for (p,q)-analogue of Meyer-K?nig and Zeller operators denoted by Mn,p,q for 0 < q < p ? 1 are obtained. Also we investigate classical and statistical versions of Korovkin type approximation results based on proposed operator. Furthermore, some graphical examples for convergence of the operators are presented.

Approximation by Modified Meyer–König and Zeller Operators via Power Series Summability Method

In this paper, we study the Korovkin-type theorem for modified Meyer–Konig and Zeller operators via A-statistical convergence and power series summability method. The rate of convergence for this new summability methods is also obtained with the help of the modulus of continuity. Further, we establish Voronovskaya-type and Gruss–Voronovskaya-type theorems for A-statistical convergence.

Product of deferred Cesàro and deferred weighted statistical probability convergence and its applications to Korovkin-type theorems

In the present work, we introduce and study the notion of statistical probability convergence for sequences of random variables as well as the idea of statistical convergence for sequences of real numbers, which are defined over a Banach space via the product of deferred Cesàro and deferred weighted summability means. We first establish a theorem presenting aconnection between them. Based upon our proposed methods, we then prove a Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space, and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results (in classical as well as in statistical versions). Furthermore, an illustrative example is presented here by means of the generalized Meyer–König and Zeller operators of a sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions. Finally, we estimate the rate of the product of deferred Cesàro and deferred weighted statistical probability convergence, and accordingly establish a new result.

Statistical probability convergence via the deferred Nörlund mean and its applications to approximation theorems

The notion of deferred weighted statistical probability convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the deferred weighted statistical convergence. Such concepts were introduced and studied by Srivastava et al. (Appl Anal Discrete Math, 2020). In the present work, we introduced and studied the notion of statistical probability convergence as well as statistical convergence for sequences of random variables and sequences of real numbers respectively defined over a Banach space via deferred Norlund summability mean. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space and demonstrated that our theorem effectively extends and improves most (if not all) of the previously existing results (in statistical versions). Finally, an illustrative example is presented here by the generalized Meyer-Konig and Zeller operators of a sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

Estimates for the Differences of Certain Positive Linear Operators

The present paper deals with estimates for differences of certain positive linear operators defined on bounded or unbounded intervals. Our approach involves Baskakov type operators, the kth order Kantorovich modification of the Baskakov operators, the discrete operators associated with Baskakov operators, Meyer–König and Zeller operators and Bleimann–Butzer–Hahn operators. Furthermore, the estimates in quantitative form of the differences of Baskakov operators and their derivatives in terms of first modulus of continuity are obtained.

Korovkin type Approximation of Abel transforms of q-Meyer-König and Zeller operators

In this paper we investigate some Korovkin type approximation properties of the $q$-Meyer-Konig and Zeller operators and Durrmeyer variant of the $q$-Meyer-Konig and Zeller operators via Abel summability method which is a sequence-to-function transformation and which extends the ordinary convergence. We show that the approximation results obtained in this paper are more general than some previous results. We also obtain the rate of Abel convergence for the corresponding operators. Finally, we conclude our results with some graphical analysis.

Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions.

Approximation by Complex Meyer-König and Zeller Operators

The Meyer-Konig and Zeller operator is one of the most challenging operators. Sometimes the study of its properties will rely on the weighted approximation by Baskakov operator. In this paper, this relation is extended to complex space; the quantitative estimates and the Voronovskaja type results for analytic functions by complex Meyer-Konig and Zeller operators were obtained.

Weighted approximation by Kanotrovich type modification of Meyer-König and Zeller operator

We investigate the weighted approximation of functions in $L_p$-norm by Kantorovich modifications of the classical Meyer-König and Zeller operator, with weights of type $(1-x)^\alpha, \alpha \in \mathbb{R}$. By defining an appropriate K-functional we prove direct theorems for them.

A Direct Theorem for MKZ-Kantorovich Operator

We characterize the approximation of functions in Lp norm by Kantorovich modification of the classical Meyer-Konig and Zeller operator. By defining an appropriate K-functional we prove a direct theorem for it.

On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems

Prior to investigating on sequence spaces and their convergence, we study the notion of statistical convergence of difference sequences of fractional order α ∈ ℝ. As generalizations of previous works, this study includes several special cases under different limiting conditions of α, such as the notion of statistical convergence of difference sequences of zeroth and mth (integer) order. In fact, we study certain new results on statistical convergence via the difference operator Δα and interpret them to those of previous works. Also, by using the convergence of Δα-summable sequences which is stronger than statistical convergence of difference sequences, we apply classical Bernstein operator and a generalized form of Meyer-Konig and Zeller operator to construct an example in support of our result. Also, we study the rates of Δα-statistical convergence of positive linear operators.

Relatively Uniform Weighted Summability Based on Fractional-Order Difference Operator

In the present paper, we introduce the notion of relatively uniform weighted summability and its statistical version based upon fractional-order difference operators of functions. The concept of relatively uniform weighted $$\alpha \beta $$ -statistical convergence is also introduced and some inclusion relations concerning the newly proposed methods are derived. As an application, we prove a general Korovkin-type approximation theorem for functions of two variables and also construct an illustrative example by the help of generating function type non-tensor Meyer-Konig and Zeller operators. Moreover, it is shown that the proposed methods are non-trivial generalizations of relatively uniform convergence which includes a scale function. We estimate the rate of convergence of approximating positive linear operators by means of the modulus of continuity and give a Voronovskaja-type approximation theorem. Finally, we present some computational results and geometrical interpretations to illustrate some of our approximation results.

Generalized Statistically Almost Convergence Based on the Difference Operator which Includes the (p, q)-Gamma Function and Related Approximation Theorems

This paper is devoted to extend the notion of almost convergence and its statistical forms with respect to the difference operator involving (p, q)-gamma function and an increasing sequence $$(\lambda _n)$$ of positive numbers. We firstly introduce some new concepts of almost $${\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )$$ -statistical convergence, statistical almost $${\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )$$ -convergence and strong almost $$[{\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )]_r$$ -convergence. Moreover, we present some inclusion relations between these newly proposed methods and give some counterexamples to show that these are non-trivial generalizations of existing literature on this topic. We then prove a Korovkin type approximation theorem for functions of two variables through statistically almost $${\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )$$ -convergence and also present an illustrative example via bivariate non-tensor type Meyer–Konig and Zeller generalization of Bernstein power series. Furthermore, we estimate the rate of almost convergence of approximating linear operators by means of the modulus of continuity and derive some Voronovskaja type results by using the generalized Meyer–Konig and Zeller operators. Finally, some computational and geometrical interpretations for the convergence of operators to a function are presented.

An Elementary Function Representation of the Second-Order Moment of the Meyer–König and Zeller Operators

We prove that the second-order moment of the Meyer–Konig and Zeller operators is an elementary function and find sharp forms of the related Becker and Nessel inequalities.

Weighted Approximation of Functions by Meyer–König and Zeller Operators of Max-Product Type

ABSTRACTIn this paper, we study the uniform approximation of functions by Meyer–König and Zeller operators of max-product type in some weighted spaces. We estimate the rate of approximation in terms of a suitable modulus of continuity.