Jakimovski-Leviatan Operators Research Articles

Unveiling the Potential of Sheffer Polynomials: Exploring Approximation Features with Jakimovski–Leviatan Operators

In this article, we explore the construction of Jakimovski–Leviatan operators for Durrmeyer-type approximation using Sheffer polynomials. Constructing positive linear operators for Sheffer polynomials enables us to analyze their approximation properties, including weighted approximations and convergence rates. The application of approximation theory has earned significant attention from scholars globally, particularly in the fields of engineering and mathematics. The investigation of these approximation properties and their characteristics holds considerable importance in these disciplines.

Confluent Appell polynomials

In this paper, we introduce the confluent Appell polynomials and prove a Sheffer type characterization theorem for them by means of the Stieltjes integral of hypergeometric polynomials. We investigate their several properties such as explicit representation, integral representation and finite summation formulas. Moreover, by proving a pure recurrence relation and deriving the lowering and the raising operators, in terms of differential and shift operators, we obtain the equation satisfied the confluent Appell polynomials by using the factorization method. And then, we define the confluent Bernoulli and Hermite polynomials and exhibit their main properties such as explicit representations, recurrence formulas (involving the corresponding usual Bernoulli and Hermite polynomials), finite summation formulas and equations involving differential and shift operators. Finally, we construct approximation operators by using confluent Appell polynomials which helps to approximate to a function defined on the semi infinite interval in a weighted function space. We call these as the confluent Jakimovski–Leviatan operators which includes the confluent version of the well-known Szász–Mirakyan operators. Also, an illustrative example in order to show convergence efficiency of the confluent Szász–Mirakyan operators is given.

A sequence of Appell polynomials and the associated Jakimovski–Leviatan operators

We are concerned with a special sequence of Appell polynomials, related to the Renyi and Tsallis entropies for the binomial distribution. The generating function is investigated: it is logarithmically convex and has remarkable connections with the modified Bessel function $$I_0(t)$$ and with the index of coincidence for Poisson distribution. The specific form of the Appell polynomials leads to specific properties of the associated Jakimovski–Leviatan operators.

Generalization of Sz\xe1sz\u2013Mirakjan\u2013Kantorovich operators using multiple Appell polynomials

The purpose of the present paper is to introduce and study a sequence of positive linear operators defined on suitable spaces of measurable functions on [0,infty ) and continuous function spaces with polynomial weights. These operators are Kantorovich type generalization of Jakimovski–Leviatan operators based on multiple Appell polynomials. Using these operators, we approximate suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,infty ) that do not constitute a subdivision of it. We also discuss the rate of convergence of these operators using moduli of smoothness.

Approximation properties of Jain-Appell operators

In this paper, we introduce the Jain-Appell operators by applying Gamma transform to the Jakimovski-Leviatan operators. In their special cases they include not only the Jain-Pethe operators, but also new families of operators, where we call them Appell-Baskakov and Appell-Lupa? operators, since their special cases contain Baskakov and Lupa? operators, respectively. We investigate their weighted approximation properties and compute the error of approximation by using certain Lipschitz class functions. Furthermore, we obtain their A-statistical approximation property.

Note On Jakimovski-Leviatan Operators Preserving e –x

Abstract In the present article, a modification of Jakimovski-Leviatan operators is presented which reproduce constant and e –x functions. We prove uniform convergence order of a quantitative estimate for the modified operators. We also give a quantitative Voronovskya type theorem.

Jakimovski–Leviatan operators of Kantorovich type involving multiple Appell polynomials

Abstract The purpose of the present paper is to obtain the degree of approximation in terms of a Lipschitz type maximal function for the Kantorovich type modification of Jakimovski–Leviatan operators based on multiple Appell polynomials. Also, we study the rate of approximation of these operators in a weighted space of polynomial growth and for functions having a derivative of bounded variation. A Voronvskaja type theorem is obtained. Further, we illustrate the convergence of these operators for certain functions through tables and figures using the Maple algorithm and, by a numerical example, we show that our Kantorovich type operator involving multiple Appell polynomials yields a better rate of convergence than the Durrmeyer type Jakimovski Leviatan operators based on Appell polynomials introduced by Karaisa (2016).

Approximation by a generalization of the Jakimovski-Leviatan operators

In this paper, we introduce a Kantorovich type generalization of Jakimovski-Leviatan operators constructed by A. Jakimovski and D. Leviatan (1969) and the theorems on convergence and the degree of convergence are established. Furthermore, we study the convergence of these operators in a weighted space of functions on [0,1).

Quantitative Voronovskaja and Grüss Voronovskaja-Type Theorems for Operators of Kantorovich Type Involving Multiple Appell Polynomials

The purpose of the present paper is to obtain the quantitative Voronovskaja and Gruss Voronovskaja-type theorems by calculating the sixth-order central moment for the Jakimovski–Leviatan operators of Kantorovich type based on multiple Appell polynomials. Also, we study the rate of approximation for functions in a Lipschitz-type space and in terms of the second-order modulus of continuity and Ditzian–Totik modulus of smoothness for continuous functions in the semi-positive axis.

Jakimovski-Leviatan operators of Durrmeyer type involving Appell polynomials

The purpose of the present paper is to establish the rate of convergence for a Lipschitz-type space and obtain the degree of approximation in terms of Lipschitz-type maximal function for the Durrmeyer type modification of Jakimovski-Leviatan operators based on Appell polynomials. We also study the rate of approximation of these operators in a weighted space of polynomial growth and for functions having a derivative of bounded variation.

Approximation by Jakimovski–Leviatan operators of Durrmeyer type involving multiple Appell polynomials

In the present paper, we introduce Jakimovski–Leviatan–Durrmeyer type operators involving multiple Appell polynomial. First, we investigate Korovkin type approximation theorem and rate of convergence by using usual modulus of continuity and class of Lipschitz function. Next, we study the convergence of these operators in weighted space of functions and estimate the approximation properties. We have also established Voronovskaja type asymptotic formula. Furthermore, we obtain statistical approximation properties of these operators with the help of universal Korovkin type statistical approximation theorem. Some graphical examples for the convergence of our operators towards some functions are given. At the end, we have computed error estimation as our numerical example.

Approximation by q-analogue of Jakimovski–Leviatan operators involving q-Appell polynomials

In the present paper, we introduce q-analogue of the Jakimovski–Leviatan operators with the help of q-Appell polynomials. We establish some moments and auxiliary results by using q-derivatives and then prove a basic convergence theorem. Also, the Voronovskaja-type asymptotic formula and some direct results for the above operators are discussed. Moreover, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied.

Rate of convergence of Chlodowsky type Durrmeyer Jakimovski-Leviatan operators

In this paper, we determine the rate of pointwise convergence of the Chlodowsky type Durrmeyer Jakimovski-Leviatan operators $L^{\ast}_{n}(f,x)$ for functions of bounded variation. We use some methods and techniques of probability theory to prove our main result.

Durrmeyer variant of q-Favard-Szász operators based on Appell polynomials

Karaisa [Karaisa, A., Approximation by Durrmeyer type Jakimoski Leviatan operators, Math. Method. Appl. Sci., DOI: 10.1002/mma.3650 (2015)] introduced the Durrmeyer type variant of Jakimovski-Leviatan operators based on Appell polynomials and studied some approximation properties. The aim of the present paper is to define the q analogue of these operators and establish the rate of convergence for a Lipschitz type space and a Lipschitz type maximal function for the Durrmeyer type variant of these operators. Also, we study the degree of approximation of these operators in a weighted space of polynomial growth and by means of weighted modulus of continuity

Approximation by modified integral type Jakimovski-Leviatan operators

In this paper, we give a generalization of the integral type Jakimovski-Leviatan operators, introduced by Ciupa [1]. Theorems on convergence and the rate of convergence of the operators by using the first and second order modulus of continuity are established. We also study the convergence of these operators in a weighted space of functions.

Approximation by Chlodowsky type Jakimovski–Leviatan operators

We introduce a generalization of the Jakimovski–Leviatan operators constructed by A. Jakimovski and D. Leviatan (1969) in [1] and the theorems on convergence and the degree of convergence are established. We also give a Voronovskaya-type theorem. Furthermore, we study the convergence of these operators in a weighted space of functions on a positive semi-axis and estimate the approximation by using a new type of weighted modulus of continuity introduced by A.D. Gadjiev and A. Aral (2007) in [9].