Summation-integral Type Operators Research Articles

Parametric Extension of a Certain Family of Summation-Integral Type Operators

In this paper, we introduce a parametric extension of a certain family of summation-integral type operators on the interval [0,∞). Firstly, we obtain test functions and central moments. Secondly, we investigate weighted approximation properties for these operators and estimate the rate of convergence. Then, we give a pointwise approximation for the Peetre K-functional and functions of the Lipschitz class. Moreover, we demonstrate Voronovskaja type theorem for the operators. Finally, the convergence properties of operators to some functions are illustrated by graphics.

Approximation by Apostol-Genocchi summation-integral type operators

Approximation by Apostol-Genocchi summation-integral type operators

Approximation Properties of Some Modified Szász–Mirakjan–Kantorovich Operators

The present article deals with approximation results obtained by means of the Lipschitz maximal function, the Ditzian–Totik modulus of smoothness, and a Lipschitz-type space having two parameters for the summation-integral type operators defined by Mishra and Yadav [22]. Further, we determine the rate of convergence in terms of a derivative of bounded variation. To estimate the quantitative results of the defined operators, we establish quantitative Voronovskaya-type and Grüss-type theorems. Moreover, examples are given with graphical representation to support the main results.

Bézier variant of summation-integral type operators

Bézier variant of summation-integral type operators

Statistical convergence of Szász-Mirakjan-Kantorovich-type operators and their bivariate extension

Our main aim is to investigate the approximation properties of the summation integral type operators in a statistical sense. In this regard, we prove the statistical convergence theorem using well known Korovkin theorem and the degree of approximation is determined. Also using the weight function, the weighted statistical convergence theorem with the help of the Korovkin theorem is obtained. The statistical rate of convergence in the terms of modulus of continuity and function belonging to the Lipschitz class is determined. To support the convergence results of the proposed operators to the function, graphical representations take place and a comparison is shown with Sz?sz-Mirakjan-Kantorovich operators through examples. The last section deals with, a bivariate extension of the proposed operators to determine the approximation of the function of two variables, additionally, the rate of convergence is estimated as well as the convergence of the bivariate operators is shown by graphical representations.

Approximation properties by some modified Szasz-Mirakjan-Kantorovich operators

The present article deals with the local approximation results by means of Lipschitz maximal function, Ditzian-Totik modulus of smoothness and Lipschitz type space having two parameters for the summation-integral type operators defined by Mishra and Yadav (Tbilisi Mathematical Journal. 11(3), (2018), 175-91). Further, we determine the rate of convergence in the term of the with derivative of bounded variation and for the quantitative means of the defined operators, we establish the quantitative Voronovskaya type and Gr$\ddot{\text{u}}$ss type theorems. Moreover, the examples are given with graphical representation to support the main results.

Approximation by a family of summation-integral type operators preserving linear functions

This article investigates the approximation properties of a general family of positive linear operators defined on the unbounded interval [0,?). We prove uniform convergence theorem and Voronovskayatype theorem for functions with polynomial growth. More precisely, we study weighted approximation i.e basic convergence theorems, quantitative Voronovskaya-asymptotic theorems and Gr?ss Voronovskayatype theorems in weighted spaces. Finally, we obtain the rate of convergence of these operators via a suitable weighted modulus of continuity.

Summation-Integral Type Operators Based on Lupas-Jain Functions

We introduce a genuine summation-integral type operators based on Lupaş-Jain type base functions related to the unbounded sequences. We investigated their degree of approximation in terms of modulus of continuity and ????-functional for the functions from bounded and continuous functions space. Furthermore, we give some theorems for the local approximation properties of functions belonging to Lipschitz class. Also, we give Voronovskaja theorem for these operators.

Iterative combinations for Srivastava–Gupta operators

In the year 2003, Srivastava–Gupta proposed a general family of linear positive operators, having some well-known operators as special cases. They investigated and established the rate of convergence of these operators for bounded variations. In the last decade for modified form of Srivastava–Gupta operators, several other generalizations also have been discussed. In this paper, we discuss the generalized modified Srivastava–Gupta operators considered in [H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling 37(12–13) (2003) 1307–1315], by using iterative combinations in ordinary and simultaneous approximation. We may have better approximation in higher order of modulus of continuity for these operators.

On approximation properties of Gupta-type operators

Summation-integral type operators play an important role in the theory of approximation concerning linear positive operators. In this direction, Gupta proposed several examples of summation-integral type operators. In one of his recent papers, some examples of genuine operators have been proposed. We discuss certain approximation properties of such operators and estimate direct results along with a Gruss-type Voronovskaya result. We obtain moments by deriving a recurrence relation and establish the difference of the genuine operators with the Mastroianni operators.

APPROXIMATION BY STANCU TYPE JAKIMOVSKI-LEVIATAN-PALTANEA OPERATORS

The present article deals with the general family of summation-integral type operators. Here, we introduce the Stancu type generalization of the Jakimovski-Leviatan- Paltanea operators and study Voronovskaja-type asymptotic theorem, local approximation, weighted approximation, rate of convergence and pointwise estimates using the Lipschitz type maximal function. Also, we propose a king type modi cation of these operators to obtain better estimates.

Blending type approximation by bézier-summation-integral type operators

In this note we construct the B€zier variant of summation integral type operators based on a non-negative real parameter. We present a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation. In the last section, we study the quantitative Voronovska ja type theorem

Better Error Estimation for the mixed Szász Baskakov type operators

This paper introduced a generalization of mixed summation integral type operators with Szász and Baskakov Basis, so-called Szász Baskakov operator. Then the moment estimates of these operators have been obtained and the uniform convergence has been established. Further, the quantitative approximation and local approximation behavior of the operators has been studied using modulus of continuity and Lipschitz class function. Then, it has been proved that the rate of convergence of the proposed operators is better than their primitives. In the last section, r−th order generalization of modified operators has been introduced and their rate of convergence has been estimated.

On approximation properties of Baskakov–Szász–Stancu operators using hypergeometric representation

In the present paper, we study the mixed summation integral type operators having Baskakov and Szász basis functions in summation and integration form, respectively. We also give here the alternate form of the operators in terms of hypergeometric functions and estimated moments of these operators using hypergeometric series. In the last section, we present some results for operators related to convergence.

Direct estimations of new generalized Baskakov-Szász operators

Several modifications of the discrete operators are available in the literature. In the recent years, certain modifications of the well-known Baskakov and Szasz-Mirakyan operators have been discussed based on certain parameters. We propose mixed summation-integral type operators and estimate the quantitative asymptotic formula and a global direct result for the special case. For general case, we establish moments and some direct convergence results in ordinary approximation, which includes point wise approximation, asymptotic formula and a direct result in terms of modulus of continuity.

Some approximation properties for generalized Srivastava– Gupta operators

The present paper deals with a sequence of modified genuine summation-integral type operators, which is a generalization to the work done by Yadav (2014) [18]. In ordinary approximation, we estimate a global result, asymptotic formula of Voronovskaja kind, error estimation in terms of modulus of continuity and also study weighted approximation for the case c∈N∪{0}, which are compared graphically using MATLAB. Further, we discuss the rate of convergence for c=−1.

Approximation for genuine summation-integral type link operators

In the present article, we extend the studies on recently introduced sequence of the genuine summation-integral type operators [7]. Here, we establish a link between the genuine operators and the discrete operators. We also establish a quantitative asymptotic formula, a direct estimate in terms of Ditzian–Totik modulus of smoothness and finally, we present the rate of convergence for functions having derivatives of bounded variation.

Approximation properties of Bezier-summation-integral type operators based on Polya–Bernstein functions

In this article we introduce the Bezier variant of summation integral type operators having Polya and Bernstein basis functions. We give a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation.

A NOTE ON THE APPROXIMATION BY THE $q$-HYBRID SUMMATION INTEGRAL TYPE OPERATORS

In this study, it is introduced a $q-$Stancu type of hybrid summation integral type operators. It is investigated their approximation properties. It is given a weighted approximation theorem and obtained rates of convergence of these operators for continuous functions.

Lupaş-Durrmeyer operators based on Polya distribution

The generalization of the Bernstein polynomials based on Polya distribution is considered in the present article. Here, we introduce a mixed summation-integral type operators having Polya and Bernstein basis functions in summation and integration respectively. We establish some direct results which include an asymptotic formula, local and global approximation results for these operators in terms of modulus of continuity.