Sequence Of Positive Linear Operators Research Articles

A Sequence of Kantorovich-Type Operators on Mobile Intervals

In this paper, we introduce and study a new sequence of positive linear operators, acting on both spaces of continuous functions as well as spaces of integrable functions on $[0, 1]$. We state some qualitative properties of this sequence and we prove that it is an approximation process both in $C([0, 1])$ and in $L^p([0, 1])$, also providing some estimates of the rate of convergence. Moreover, we determine an asymptotic formula and, as an application, we prove that certain iterates of the operators converge, both in $C([0, 1])$ and, in some cases, in $L^p([0, 1])$, to a limit semigroup. Finally, we show that our operators, under suitable hypotheses, perform better than other existing ones in the literature.

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 Sequences of J. P. King-Type Operators

This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions andx2on[0,1]. Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend King’s approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the Szász-Mirakyan operators.

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.

Approximation by a sequence of operators of the Srivastava‐Gupta type involving the Brenke polynomials with a certain parameter

We introduce a new sequence of linear positive operators by combining the Brenke polynomials and the Srivastava‐Gupta–type operators defined by Srivastava‐Gupta. obtain the moments of the operators and present some classical and statistical approximation properties by means of Korovkin results. Next, we estimate a global result, which includes the Voronovskaya‐type asymptotic formula, local approximation, error estimation in terms of weighted modulus of continuity, and for functions in a Lipschitz‐type space. Lastly, we estimate the rate of approximation for functions with derivatives of bounded variation.

Approximation by genuine Gupta–Srivastava operators

In the present paper, we consider new operators, which is defined by Gupta and Srivastava (Eur J Pure Appl Math 11(3):575–579, 2018). They considered a general sequence of positive linear operators and gave the modified form of their previous operators (Neer et al. in Math Comput Model 37:1307–1315, 2003). As these operators preserve linear functions, we call these operators as genuine Gupta–Srivastava operators. Here we discuss some basic properties, direct results and rate of convergence of functions of bounded variation and weighted approximation.

Quantitative Voronovskaya type theorems for a general sequence of linear positive operators

The present paper deal with the obtaining quantitative form of the results presented Butzer & Karsli [1]. That is, we prove quantitative simultaneous results by general sequence of positive linear operators which are valid for unbounded functions with polynomial growth. We present some applications of the general results by considering particular sequences of positive linear operators.

A sequence of positive linear operators related to powered Baskakov basis

In this paper we study some approximation properties of a sequence of positive linear operators defined by means of the powered Baskakov basis. We prove that in the particular case of squared Baskakov basis the operators behave better than the classical Baskakov operators. For this particular case we give also a quantitative Voronovskaya type result.

Relative weighted almost convergence based on fractional-order difference operator in multivariate modular function spaces

In the present paper, we introduce the concept of relative weighted almost convergence and its weighted statistical extensions in multivariate modular function spaces based on a new type fractional-order double difference operator $$\nabla ^{a,b,c}_h$$ . We first define the concepts of weighted almost $$\nabla $$ -statistical convergence and statistical weighted almost $$\nabla $$ -convergence of double sequences. By using the notion of relative uniform convergence involving a scale function $$\sigma $$ , we introduce modular relative weighted almost statistical convergence and modular relative statistical weighted almost convergence of double sequences. We then obtain some inclusion relations between these proposed methods and provide some counterexamples that show that these are non-trivial and proper extensions of the existing literature on this topic. Moreover, we apply the relative statistical weighted almost convergence of a double sequence of positive linear operators to prove some Korovkin-type theorems in multivariate modular spaces by considering several kinds of test functions. Finally, we present a non-trivial application to generalized Boolean sum (GBS) operators of bivariate generalized Bernstein–Durrmeyer operators.

The Approximation of Bivariate Blending Variant Szász Operators Based Brenke Type Polynomials

We have constructed a new sequence of positive linear operators with two variables by using Szasz-Kantorovich-Chlodowsky operators and Brenke polynomials. We give some inequalities for the operators by means of partial and full modulus of continuity and obtain a Lipschitz type theorem. Furthermore, we study the convergence of Szasz-Kantorovich-Chlodowsky-Brenke operators in weighted space of function with two variables and estimate the rate of approximation in terms of the weighted modulus of continuity.

KOROVKIN THEOREM VIA STATISTICAL e-MODULAR CONVERGENCE OF DOUBLE SEQUENCES

In the present paper, we obtain an abstract version of the Korovkin type theorem via the concept of statistical e-convergence in modular spaces for double sequences of positive linear operators. We give an application showing that the new result is stronger than classical ones. Also, we study an extension to non-positive operators.

Voronovskaya theorem for a sequence of positive linear operators related to squared Bernstein polynomials

In this paper we obtain the Voronovskaya formula for the sequence of positive linear operators constructed using the squared Bernstein polynomials.

Szász Type Operators Involving Charlier Polynomials of Blending Type

We introduce a new sequence of linear positive operators by combining the Charlier polynomials and the Szasz type operators defined by Jain (in J Aust Math Soc 13(3):271–276, 1972). We obtain the degree of approximation of these operators by means of the weighted modulus of continuity and for functions in a Lipschitz type space. We also study the approximation of functions having a derivative equivalent with a function of bounded variation.

Abstract Korovkin Theory in Modular Spaces in The Sense of Power Series Method

In this paper, using the power series method we obtain an abstract Korovkin type approximation theorem for a sequence of positive linear operators defined on modular spaces.

Approximation of General Form for a Sequence of Linear Positive Operators Based on Four Parameters

In the present paper, we define a generalization sequence of linear positive operators based on four parameters which is reduce to many other sequences of summation–integral older type operators of any weight function (Bernstein, Baskakov, Szász or Beta). Firstly, we find a recurrence relation of the -th order moment and study the convergence theorem for this generalization sequence. Secondly, we give a Voronovaskaja-type asymptotic formula for simultaneous approximation. Finally, we introduce some numerical examples to view the effect of the four parameters of this sequence.

Asymptotic Evaluations for Some Sequences of Positive Linear Operators

We prove asymptotic evaluations for univariate and multivariate positive linear operators. Our proofs are different from what has been used so far. As applications of our results, we find the full asymptotic evaluation for the iterates of the univariate Cesaro and Volterra operators. Moreover, we find asymptotic evaluations for the iterates of multivariate Cesaro and Volterra type operators on the k-dimensional unit cube, k-dimensional unit triangle, etc.

Approximation by Baskakov-Durrmeyer operators based on (p,q)-integers

AbstractIn the present paper, we introduce a new sequence of linear positive operators based on (p,q)-integers. To approximate functions over unbounded intervals, we introduce Baskakov-Durrmeyer type operators using the (p,q)-Gamma function. We investigate rate of convergence of new operators in terms of modulus of continuities and obtain their approximation behavior for the functions belonging to Lipschitz class. At the end, we present a modification of new operators preserving the test functionx.

Approximation properties of \u03bb-Kantorovich operators

In the present paper, we study a new type of Bernstein operators depending on the parameter lambdain[-1,1]. The Kantorovich modification of these sequences of linear positive operators will be considered. A quantitative Voronovskaja type theorem by means of Ditzian–Totik modulus of smoothness is proved. Also, a Grüss–Voronovskaja type theorem for λ-Kantorovich operators is provided. Some numerical examples which show the relevance of the results are given.

A General Family of the Srivastava-Gupta Operators Preserving Linear Functions

The general sequence of positive linear operators containing some well-known operators as special cases were introduced in the earlier work by Srivastava and Gupta [9], which reproduce only the constant functions. In the present sequel, we provide a general sequence of operators which preserve not only the constant functions, but also linear functions.

Statistical equi-equal convergence of positive linear operators

Many researchers have been interested in the concept of statistical convergence because of the fact that it is stronger than the classical convergence. Also, the concepts of statistical equal convergence and equi-statistical convergence are more general than the statistical uniform convergence. In this paper we define a new type of statistical convergence by using the notions of equi-statistical convergence and statistical equal convergence to prove a Korovkin type theorem. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were demonstrated by earlier authors. After, we present an example in support of our definition and result presented in this paper. Finally, we also compute the rates of statistical equi-equal convergence of sequences of positive linear operators.