Exponential-type Operators Research Articles

Estimation using a summation integral operator of exponential type with a weight derived from the α‐Baskakov basis function

With the weight of ‐Baskakov basis function, this article considers the approximation properties of the integral modification of exponential type operators. The moments are estimated, and the Voronovskaja‐type result is established in simultaneous approximation. Additionally, the power series method is used for estimating convergence and error estimation. For some functions, the convergence of these operators has also been graphically illustrated.

Approximation by a modification of operators of exponential type associated with the Baskakov operators

In the current article, we modify the operators of exponential type associated with the Baskakov operators so as to preserve the linear functions. Initially, we obtain the moments and central moments for the modified form. Further, we derive few convergence results including Voronovskaja type asymptotic formula and validate our results through graphical illustration. In the end, we obtain the difference estimate between the exponential type operators associated with the Baskakov operators and its modified form.

Approximation properties of exponential type operators connected to $ p(x) = 2x^{3/2} $

In this paper, we provide a state-of-the-art survey of some recent results regarding approximation properties of exponential type operators connected with $ 2x^{3/2} $ obtained by professor Gupta and his collaborators in the last few years. The pioneer work in introduction and study of exponential type operators was due to May and Ismail, earlier, in 1976 and 1978. Gupta, generalized and deepened this study through his many works in recent years. We bring together in discussion and give all existence theorems for this specific operators and their modification and present all their approximation properties that was studied until now. In the final section, the theoretical results are analyzed by numerical examples.

On new exponential-type operators

On new exponential-type operators

Some new semi-exponential operators

In the theory of approximation, linear operators play an important role. The exponential-type operators were introduced four decades ago, since then no new exponential-type operator was introduced by researchers, although several generalizations of existing exponential-type operators were proposed and studied. Very recently, the concept of semi-exponential operators was introduced and few semi-exponential operators were captured from the exponential-type operators. It is more difficult to obtain semi-exponential operators than the corresponding exponential-type operators. In this paper, we extend the studies and define semi-exponential Bernstein, semi-exponential Baskakov operators, semi-exponential Ismail–May operators related to 2x^{3/2} or x^{3}. Furthermore, we present a new derivation for the semi-exponential Post–Widder operators. In some examples, open problems are indicated.

Voronovskaya-type Results for Positive Linear Operators of Exponential Type and their Derivatives

Voronovskaya-type Results for Positive Linear Operators of Exponential Type and their Derivatives

Weighted A-statistical convergence and Bögel approximation by operators of exponential type

Weighted A-statistical convergence and Bögel approximation by operators of exponential type

Modification of Exponential Type Operators Preserving Exponential Functions Connected with $$x^3$$

Modification of Exponential Type Operators Preserving Exponential Functions Connected with $$x^3$$

Semi-Exponential Operators

In this paper we study approximation properties of exponential-type operators for functions from exponential weighted spaces. We focus on some modifications of these operators and we derive a new example of such operators. A probabilistic approach for these modifications is also demonstrated.

Higher order Lupaş-Kantorovich operators and finite differences

It is well-known that Lupas operators are not exponential type operators, although they preserve linear functions. The differential operator is not applicable to estimate nice representation between Lupas operators and its Kantorovich variant. In the present paper we use finite difference method viz. backward differences to estimate the recurrence relation for moments of the Lupas operators. We introduce higher order Lupas-Kantorovich operators and estimate some direct results.

On Modification of Certain Exponential Type Operators Preserving Constant and $$e^{-x}$$

The key goal of this article is to propose a modification of certain exponential type operators defined by Ismail and May. Particularly, we concentrate on a sequence of operators that preserve $$e^{-x}$$ and constant functions. We find the moments of these modified operators using the concept of moment generating function with the help of Mathematica software. We show uniform convergence of these modified operators and analyze the asymptotic behaviour with a Voronovskaya type theorem. We also illustrate via graphs that our modified operators approximate better than the original operators for certain family of functions. Finally, we show the convergence of these modified operators graphically using Mathematica Software.

Asymptotic expansions for certain exponential-type operators connected with 2x^{3/2}

In the present paper, we consider the complete asymptotic expansion of certain exponential-type operators connected with 2x^{3/2}. Also, a modification of such exponential-type operators is provided, which preserve the function mathrm{e}^{Ax}.

Approximation by Some Baskakov–Kantorovich Exponential-Type Operators

In the present paper, we propose the modification of the Baskakov–Kantorovich operators based on $$\mu $$ -integral. Such operators are connected with exponential functions. We estimate moments and establish some direct results in terms of modulus of continuity.

A complete asymptotic expansion for operators of exponential type with pleft( xright) =xleft( 1+xright) ^{2}

In the year 1978, Ismail and May studied operators of exponential type and proposed some new operators which are connected with a certain characteristic function pleft( xright) . Several of these operators were not separately studied by researchers due to its unusual behavior. The topic of the present paper is the local rate of approximation of a sequence of exponential type operators R_{n} belonging to pleft( xright) =xleft( 1+xright) ^{2}. As the main result we derive a complete asymptotic expansion for the sequence left( R_{n}fright) left( xright) as n tends to infinity.

Rate of convergence of exponential type operators related to pleft( xright) =2x^{3/2} for functions of bounded variation

The present article deals with the approximation of certain exponential type operators defined by Ismail and May. We estimate the rate of convergence of these operators for functions of bounded variation.

Approximation for modification of exponential type operators connected with $$x(x+1)^2$$

In the present paper, we discuss certain approximation properties of the modification of exponential type operators connected with $$x(x+1)^2$$ . In modification, we consider the weights of Baskakov basis. We estimate moments and establish a Voronovskaja-kind asymptotic formula in simultaneous approximation. We also estimate the difference of modified operators with the original exponential operators. In the end the convergence of a polynomial function has been shown through graphical representation.

A Note on the Difference of Positive Operators and Numerical Aspects

Recently, the differences between the two operators get the attention of scientists in approximation theory due to their ability to provide the approximation properties of the operator in the difference if the approximation properties of other operator in the difference are known. In other words, it gives us the ability to obtain a simultaneous approximation. On the other hand, the exponential-type operators possess better approximation properties than classical ones. Herein, the differences of the exponential-type Bernstein and Bernstein–Kantorovich operators and their differences between their higher order $$\mu $$-derivatives applied to a function with the operators applied to the same order of $$\mu $$-derivative of the function are considered. The estimates in the quantitative form are given in terms of the first modulus of continuity. Furthermore, quantitative estimates of the differences between Bernstein and Bernstein–Kantorovich operators as well as their Gruss-type difference are obtained. The numerical results obtained are in the direction of the theory, and some of them are presented.

Operators of exponential type and the abstract Cauchy problem

In this paper, we introduce closed operators of exponential type, and use it to study the solution of the homogeneous abstract Cauchy problem of the first order, usual and fractional.

Convergence estimates of a family of approximation operators of exponential type

The main object of this article is to consider a family of approximation operators of exponential type, which has presumably not been studied earlier due mainly to their seemingly complicated behavior. We estimate and establish a quantitative asymptotic formula in terms of the modulus of continuity with exponential growth, a Korovkin-type result for exponential functions and also a Voronovskaja-type asymptotic formula in the simultaneous approximation.

A Voronovskaya-Type Theorem for the First Derivatives of Positive Linear Operators

In this paper we obtain a Voronovskaya formula for derivatives of positive linear operators from a general class of exponential-type operators.