Sequence Of Positive Linear Operators Research Articles

Korovkin Second Theorem via -Statistical -Summability

Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, , and in the space as well as for the functions 1, cos, and sin in the space of all continuous 2-periodic functions on the real line. In this paper, we use the notion of -statistical -summability to prove the Korovkin second approximation theorem. We also study the rate of -statistical -summability of a sequence of positive linear operators defined from into .

Korovkin-Type Theorems in WeightedLp-Spaces via Summation Process

Korovkin-type theorem which is one of the fundamental methods in approximation theory to describe uniform convergence of any sequence of positive linear operators is discussed on weighted L p spaces, 1 ≤ p < ∞ for univariate and multivariate functions, respectively. Furthermore, we obtain these types of approximation theorems by means of 𝒜-summability which is a stronger convergence method than ordinary convergence.

Q-analogue of a new sequence of linear positive operators

Abstract This paper deals with Durrmeyer type generalization of q-Baskakov type operators using the concept of q-integral, which introduces a new sequence of positive q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous functions defined on the interval [ 0 , ∞ ) . An estimate for the rate of convergence and weighted approximation properties are also obtained. MSC:41A25, 41A36.

Direct and strong converse theorems for a general sequence of positive linear operators

We present direct and strong converse theorems for a general sequence of positive linear operators satisfying some functional equations. The results can be applied to some extensions of Baskakov and Szasz–Mirakyan operators.

A-summation process and Korovkin-type approximation theorem for double sequences of positive linear operators

Abstract The aim of this paper is to present a Korovkin-type approximation theorem on the space of all continuous real valued functions on any compact subset of the real two-dimensional space by using a A-summation process. We also study the rates of convergence of positive linear operators with the help of the modulus of continuity.

On a Generalization of Szász–Mirakjan–Kantorovich Operators

In this paper we introduce and study a sequence of positive linear operators acting on suitable spaces of measurable functions on [0,+∞[, including L p ([0,+∞[) spaces, 1 ≤ p < +∞, as well as continuous function spaces with polynomial weights. These operators generalize the Szasz–Mirakjan–Kantorovich operators and they allow to approximate (or to reconstruct) suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,+∞[ that do not constitute a subdivision of it. We also give some estimates of the rates of convergence by means of suitable moduli of smoothness.

Asymptotic Formulae via a Korovkin‐Type Result

We present a sort of Korovkin‐type result that provides a tool to obtain asymptotic formulae for sequences of linear positive operators.

On Some Extensions of Szasz Operators Including Boas‐Buck‐Type Polynomials

This paper is concerned with a new sequence of linear positive operators which generalize Szasz operators including Boas‐Buck‐type polynomials. We establish a convergence theorem for these operators and give the quantitative estimation of the approximation process by using a classical approach and the second modulus of continuity. Some explicit examples of our operators involving Laguerre polynomials, Charlier polynomials, and Gould‐Hopper polynomials are given. Moreover, a Voronovskaya‐type result is obtained for the operators containing Gould‐Hopper polynomials.

Statistical approximation of a kind of Kantorovich type [formula omitted]-Szász–Mirakjan operators

In the present paper, we introduce a Kantorovich type generalization of the q -Szász–Mirakjan operators and investigate their A -statistical approximation properties. Also, the rate of A -statistical convergence of the proposed sequence of linear positive operators is obtained by a weighted modulus of continuity.

Statistical σ approximation to Bögel-type continuous functions

In this paper, using the concept of statistical σ-convergence which is stronger than the statistical convergence, we obtain a statistical σ-approximation theorem for sequences of positive linear operators defined on the space of all real valued B-continuous functions on a compact subset of the real line. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. Also we compute the rate of statistical σ-convergence of sequence of positive linear operators.

Universal properties of approximation operators

We discuss universal properties of some operators L n : C [ 0 , 1 ] → C [ 0 , 1 ] . The operators considered are closely related to a theorem of Korovkin (1960) [4] which states that a sequence of positive linear operators L n on C [ 0 , 1 ] is an approximation process if L n f i → f i ( n → ∞ ) uniformly for i = 0 , 1 , 2 , where f i ( x ) = x i . We show that L n f may diverge in a maximal way if any requirement concerning L n in this theorem is removed. There exists for example a continuous function f such that ( L n f ) n ∈ N is dense in ( C [ 0 , 1 ] , ‖ . ‖ ∞ ) , even if L n is positive, linear and satisfies L n P → P ( n → ∞ ) for all polynomials P with P ( 0 ) = 0 .

Korovkin-Type Approximation Theorem for Double Sequences of Positive Linear Operators via Statistical A-Summability

In this paper, using the concept of statistical A-summability which is stronger than the A-statistical convergence we provide a Korovkin-type approximation theorem on the space of all continuous real valued functions defined on any compact subset of the real two-dimensional space. We also study the rates of statistical A-summability of positive linear operators.

An extension based on qR-integral for a sequence of operators

Abstract The paper deals with a sequence of linear positive operators introduced via q -Calculus. We give a generalization in Kantorovich sense of its involving qR -integrals. Both for discrete operators and for integral operators we study the error of approximation for bounded functions and for functions having a polynomial growth. The main tools consist of the K -functional in Peetre sense and different moduli of smoothness.

Bernstein-type operators which preserve polynomials

In this paper we present the sequence of linear Bernstein-type operators defined for f ∈ C [ 0 , 1 ] by B n ( f ∘ τ − 1 ) ∘ τ , B n being the classical Bernstein operators and τ being any function that is continuously differentiable ∞ times on [ 0 , 1 ] , such that τ ( 0 ) = 0 , τ ( 1 ) = 1 and τ ′ ( x ) > 0 for x ∈ [ 0 , 1 ] . We investigate its shape preserving and convergence properties, as well as its asymptotic behavior and saturation. Moreover, these operators and others of King type are compared with each other and with B n . We present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improve B n in approximating a number of increasing functions, and which, apart from the constant functions, fix suitable polynomials of a prescribed degree. The notion of convexity with respect to τ plays an important role.

Localization results for generalized Baskakov/Mastroianni and composite operators

In this paper we study localization results for classical sequences of linear positive operators that are particular cases of the generalized Baskakov/Mastroianni operators and also for certain class of composite operators that can be derived from them by means of a suitable transformation. Amongst these composite operators we can find classical sequences like the Meyer–König and Zeller operators and the Bleimann, Butzer and Hahn ones. We extend in different senses the traditional form of the localization results that we find in the classical literature and we show several examples of sequences with different behavior to this respect.

Generalized Szász Durrmeyer operators

In this paper, we introduce and study a new sequence of positive linear operators acting on the spaces of continuous function on positive semi-axis. These operators are defined by means of the q-integral, which generalize the Szasz Durrmeyer operators. We study their approximation property by presenting a direct approximation theorem on polynomial weighted spaces and the rate of convergence by means of weighted modulus of continuity. Also, we establish an asymptotic formula with respect to weighted norm.

Some approximation results for Durrmeyer operators

In the present paper, we obtain a sequence of positive linear operators which has a better rate of convergence than the Szász–Mirakian Durrmeyer and Baskakov Durrmeyer operators and their Voronovskaya type results.

Approximation in statistical sense to n-variate B-continuous functions by positive linear operators

Abstract Our primary interest in the present paper is to prove a Korovkintype approximation theorem for sequences of positive linear operators defined on the space of all real valued n-variate B-continuous functions on a compact subset of the real n-dimensional space via statistical convergence. Also, we display an example such that our method of convergence is stronger than the usual convergence.

Statistical [formula omitted]-convergence of positive linear operators

Mursaleen and Edely [M. Mursaleen and O.H.H Edely, On invariant mean and statistical convergence, Appl. Math. Lett. (2009) doi:10.1016/j.aml.2009.06.005] have recently introduced the notion of statistical σ -convergence. In this paper, we study its use in the Korovkin-type approximation theorem. Then, we construct an example such that our new result works but its classical and statistical cases do not work. We also compute the rates of statistical σ -convergence of a sequence of positive linear operators.

Equi-statisticalσ-convergence of positive linear operators

In this paper we introduce the notion of equi-statistical σ -convergence which is stronger than the uniform convergence (in the ordinary sense), statistical uniform convergence and statistical uniform σ -convergence. Then, we also give its use in the Korovkin-type approximation theory. We also compute the rate of equi-statistical σ -convergence of a sequence of positive linear operators.