On Some Approximation Properties for a Sequence of λ-Bernstein Type Operators (original) (raw)

On Better Approximation of the Squared Bernstein Polynomials

Journal of University of Anbar for Pure Science

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. .

Approximation By New Family of Modified Bernstein Type Polynomials

International Journal of Innovative Technology and Exploring Engineering, 2019

In the present paper our main aim is to use the approximation methods to express the Laplace formula of theory of probability by new family of modified Bernstein Type Polynomials defined for the function f(u) of .

On approximation properties of a generalization of Bernstein polynomials in symmetric range

New Trends in Mathematical Science, 2021

In the present paper, in order to make the convergence faster to a function being approximated we identify a new generalization of Bernstein operators depending on symmetric range. The rate of convergence of these operators are given by using the modulus of continuity. Furthermore,we establish Korovkin-type approximation theorem and Voronovskaja type asymptotic theorem. Finally, we show that using graphics in Maple this new generalization of Bernstein operators converge faster than Bernstein operators on symmetric range for certain functions.

On approximation by some Bernstein–Kantorovich exponential-type polynomials

Periodica Mathematica Hungarica, 2019

Since the introduction of Bernstein operators, many authors defined and/or studied Bernstein type operators and their generalizations, among them are Morigi and Neamtu (Adv Comput Math 12:133-149, 2000). They proposed an analog of classical Bernstein operator and proved some convergence results for continuous functions. Herein, we introduce their integral extensions in Kantorovich sense by replacing the usual differential and integral operators with their more general analogues. By means of these operators, we are able to reconstruct the functions which are not necessarily continuous. It is shown that the operators form an approximation process in both C [0, 1] and L p,μ [0, 1], which is an exponentially weighted space. Also, quantitative results are stated in terms of appropriate moduli of smoothness and K-functionals. Furthermore, a quantitative Voronovskaya type result is presented.

Statistical Convergence of Bernstein Operators

Applied Mathematics & Information Sciences, 2016

The Bernstein operator is one of the important topics of approximation theory in which it has been studied in great details for a long time. The aim of this paper is to study the statistical convergence of sequence of Bernstein polynomials. In this paper, we introduce the concepts of statistical convergence of Bernstein polynomials and V B −summability and related theorems. We also study Korovkin type-convergence of Bernstein polynomials.

Asymptotic expansion and extrapolation for Bernstein polynomials with applications

BIT Numerical Mathematics, 1996

Given a real function f c C2 ' [ 0, 1], k > 1 and the corresponding Bernstein polynomials {B, (f)}, we derive an asymptotic expansion formula for & (f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence than Bn (f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique .