Sequence Of Positive Linear Operators Research Articles

The Convergence of Some Positive Linear Operators on the Space of Multivariate Continuous Periodic Functions

As a consequence of a general result, we prove that in the case of singular integrals the set of convergence consists only of the two functions 1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{1}$$\\end{document} and cos\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cos $$\\end{document}. We prove also a multivariate version of this result and apply it to find the necessary and sufficient conditions for the convergence of the sequences of positive linear operators associated to the rectangular and triangular summation.

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.

Kantorovich Stancu Type Operator Including Generalized Brenke Polynomials

This article is concerned with the sequence of operators of Stancu’s-type, involving extended Brenke polynomials. We apply Korovkin’s theorem to the sequence of positive linear operators, discuss the uniform approximation of continuous functions on closed bounded intervals by known tools theory, and also consider the second modulus of continuity, Peetre’s K-functional and Lipschitz class, which are essential concepts in approximation theory.

Some Approximation Properties of Operators Including Degenerate Appell Polynomials

ABSTRACT This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.

On Some Approximation Processes Generated by Integrated Means on Noncompact Real Intervals

The purpose of the present paper is to carry out a detailed study of a sequence of positive linear operators acting on continuous function spaces on an arbitrary real interval and constructed by means of (Borel) integrated means with respect to two families of probability Borel measures on the underlying interval and a positive real parameter. The study is mainly focused on their approximation properties in weighted spaces of continuous functions with respect to wide classes of weights. Pointwise estimates as well as weighted norm estimates are also established. In the final section a weighted asymptotic formula is obtained.

A Certain Class of Equi-Statistical Convergence in the Sense of the Deferred Power-Series Method

In this paper, we expose the ideas of point-wise statistical convergence, equi-statistical convergence and uniform statistical convergence in the sense of the deferred power-series method. We then propose a relation connecting them, which is followed by several illustrative examples. Moreover, as an application viewpoint, we establish an approximation theorem based upon our proposed method for equi-statistical convergence of sequences of positive linear operators. Finally, we estimate the equi-statistical rates of convergence for the effectiveness of the results presented in our study.

Approximation via statistical measurable convergence with respect to power series for double sequences

Abstract In this paper, we introduce and study a new type of convergences using statistical convergence via the power series method and measurable convergence. We also study their relationship with other convergences. Further, we demonstrate Korovkin-type approximation theorems for double sequences of positive linear operators using these newly specified convergences, and we also provide illustrations that demonstrate how our proven theorems are better than their classical counterparts. Finally, we have determined rates of statistical product measurable convergence using the power series approach and the modulus of continuity.

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.

An operator version of the Korovkin theorem for continuous periodic functions

The following operator version of the Korovkin theorem for continuous 2π-periodic functions is proved: Let Ln:C2π(R)→C2π(R) be a sequence of linear positive operators and U:C2π(R)→C2π(R) a linear positive operator such that [U(sin⁡)]2+[U(cos⁡)]2=[U(1)]2. If limn→∞⁡Ln(1)=U(1), limn→∞⁡Ln(sin⁡)=U(sin⁡), limn→∞⁡Ln(cos⁡)=U(cos⁡) all uniformly on R, then for every φ∈C2π(R) we have limn→∞⁡[Ln(φ)U(1)−Ln(1)U(φ)]=0 uniformly on R. If in addition, U(1)(x)>0 for every x∈R then for every φ∈C2π(R), limn→∞⁡Ln(φ)=U(φ) uniformly on R. As application we give various concrete examples.

Α-Baskakov-Durrmeyer type operators and their approximation properties

In the present research article, we construct a new family of summation-integral type hybrid operators in terms of shape parameter ? ? [0, 1]. Further, basic estimates, rate of convergence and the order of approximation with the aid of Korovkin theorem and modulus of smoothness are investigated. Moreover, numerical simulation and graphical approximations are studied. For these sequences of positive linear operators, we study the local approximation results using Peetre?s K-functional, Lipschitz class and modulus of smoothness of second order. Next, we obtain the approximation results in weighted space. Lastly, A-statistical-approximation results are presented.

Rate of convergence by Kantorovich type operators involving adjoint Bernoulli polynomials

We introduce a sequence of positive linear operators involving adjoint Bernoulli polynomials of the first kind, and we focus on the approximation properties of these operators. One of the main objectives is to get estimates for the order of approximation by means of first-order modulus of continuity, the Lipschitz condition, first modulus of derivative and a combination of first order modulus of continuity and extended second-order modulus. Further, we give Voronovskaya type and Gr?ss-Voronovskaya type asymptotic results. Finally, we give two examples for error estimation by using Maple software.

Approximation properties of a discrete operator

Abstract In the present research, we estimate some approximation properties for the generalization of a sequence of positive linear operators considered by Holhoş [6] with the following operators available in literature as special examples: Szàsz operator, Ismail-May operator associated with x(1 + x)2. We estimate its recurrence relation, moments and verify the same for its particular cases. Finally, we graphically compare the convergence of this new general operator for a continuous function, examine the convergence for each instance, and come to a conclusion regarding which scenario provides the best approximation.

On Better Approximation of the Squared Bernstein Polynomials

The present paper is defined a new better approximation of the squared Bernstein polynomials. This better approximation has been built on a positive function defined on the interval [0,1] which has some properties. First, the moderate uniform convergence theorem for a sequence of linear positive operators (the generalization of the Korovkin theorem) of these polynomials is improved. Then, the rate of convergence of these polynomials corresponding to the first and second modulus of continuity and Ditzian- Totik modulus of smoothness is given. Also, the quantitative Voronovskaja and the Grüss- Voronovskaja theorems are discussed. Finally, some numerically applied for these polynomials are given by choosing a test function and two different functions show the effect of the different chosen functions . It turns the new better approximation of the squared Bernstein polynomials gives us a better numerical result than the numerical results of both the classical Bernstein polynomials and the squared Bernstein polynomials. MSC 2010. 41A10, 41A25, 41A36.

Convergence of Special Sequences of Semi-Exponential Operators

Several papers, mainly written by J. de la Call and co-authors, contain modifications of classical sequences of positive linear operators to obtain new sequences converging to limits which are not necessarily the identity operator. Such results were obtained using probabilistic methods. Recently, results of this type have been obtained with analytic methods. Semi-exponential operators have also been introduced, extending the theory of exponential operators. We combine these two approaches, applying the semi-exponential operators in a new context and enlarging the list of operators representable as limits of other operators.

Operators Obtained by Using Certain Generating Function for Approximation

This paper is concerned with the sequence of positive linear operators obtained by certain generating functions of polynomials and with investigation of its approximation properties in detail. Initially, the convergence theorem is expressed for the sequence constructed in this article using the universal Korovkin-type theorem and then considering the modulus of continuity and the Lipschitz class, an estimate of the degree of approximation is obtained for this sequence of positive linear operators. Moreover, the generalization involving integral of these operators is defined and then their approximation properties are examined.

Korovkin-type theorems and local approximation problems

Of concern are local approximation problems for sequences of positive linear operators acting on linear subspaces of functions defined on a metric space. A Korovkin-type theorem is established in such a framework together with several consequences related to one dimensional, multidimensional and infinite dimensional settings (Hilbert spaces).Furthermore, some applications are discussed which concern classical sequences of positive linear operators including (one dimensional and multidimensional) Bernstein operators, Kantorovich operators, Szász–Mirakyan operators, Gauss–Weierstrass operators and Bernstein–Schnabl operators on convex subsets of Hilbert spaces.Finally the paper ends with a reassessment of a result of Korovkin concerning subspaces of bounded 2π− periodic functions on R and with an application related to sequences of convolution operators generated by positive approximate identities.

An operator version of the Korovkin theorem

We prove the following operator version of the Korovkin theorem: Let T be a compact Hausdorff space, Vn:C[a,b]→C(T) a sequence of linear positive operators and A:C[a,b]→C(T) a linear positive operator such that A(1)A(e2)=[A(e1)]2 and A(1)(t)>0, ∀t∈T. If limn→∞⁡Vn(1)=A(1), limn→∞⁡Vn(e1)=A(e1), limn→∞⁡Vn(e2)=A(e2) all uniformly on T, then for every f∈C[a,b], limn→∞⁡Vn(f)=A(f) uniformly on T. Many and various examples are given.

Approximation of Real Functions by a Generalization of Ismail–May Operator

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth.

On Shape Parameterα-Based Approximation Properties andq-Statistical Convergence of Baskakov-Gamma Operators

We construct a novel family of summation-integral-type hybrid operators in terms of shape parameterα∈0,1in this paper. Basic estimates, rate of convergence, and order of approximation are also studied using the Korovkin theorem and the modulus of smoothness. We investigate the local approximation findings for these sequences of positive linear operators utilising Peetre’s K-functional, Lipschitz class, and second-order modulus of smoothness. The approximation results are then obtained in weighted space. Finally, for these operatorsq-statistical convergence is also investigated.

Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful concepts. We also state and prove two Korovkin-type approximation theorems involving algebraic test functions by using our proposed concepts and methodologies. Furthermore, in order to demonstrate the usefulness of our findings, we consider an illustrative example involving a sequence of positive linear operators in conjunction with the familiar Bernstein polynomials. Finally, in the concluding section, we propose some directions for future research on this topic, which are based upon the core concept of statistical Lebesgue-measurable sequences of functions.