Bernstein-type operators which preserve polynomials (original) (raw)

Generalized Bernstein Operators on the Classical Polynomial Spaces

Mediterranean Journal of Mathematics

We study generalizations of the classical Bernstein operators on the polynomial spaces P n [a, b], where instead of fixing 1 and x, we reproduce exactly 1 and a polynomial f 1 , strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing 1 and f 1. These operators are defined by non-decreasing sequences of nodes precisely when f ′ 1 > 0 on (a, b), but even if f ′ 1 vanishes somewhere inside (a, b), they converge to the identity.

Convexity and generalized Bernstein polynomials

Proceedings of the Edinburgh Mathematical Society, 1999

In a recent generalization of the Bernstein polynomials, the approximated function f is evaluated at points spaced at intervals which are in geometric progression on [0, 1], instead of at equally spaced points. For each positive integer n, this replaces the single polynomial Bnf by a one-parameter family of polynomials , where 0 < q ≤ 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning when f is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if f is increasing then is increasing, and if f is convex then is convex, generalizing well known results when q = 1. It is also shown that if f is convex then, for any positive integer n This supplements the well known classical result that when f is convex.

Bernstein Polynomials and Operator Theory

Results in Mathematics, 2009

Kelisky and Rivlin have proved that the iterates of the Bernstein operator (of fixed order) converge to L, the operator of linear interpolation at the endpoints of the interval [0, 1]. In this paper we provide a large class of (not necessarily positive) linear bounded operators T on C[0, 1] for which the iterates T n converge towards L in the operator norm. The proof uses methods from the spectral theory of linear operators.

A Note On Bernstein-Stancu-Chlodowsky Operators

Kırıkkale Universitesi Bilimde Gelismeler Dergisi, 2012

In this note, approximation of functions by Bernstein-Stancu-Chlodowsky polynomials C n; ; (f ; x) is introduced on the interval [0; A] and weighted approximation of functions by C n; ; (f ; x) on [0; 1) by the modulus of continuity of f and f 0 are studied.

The Combination of Bernstein Polynomials with Positive Functions Based on a Positive Parameter \(s\)

Communications in Mathematics and Applications

This paper deals with a sequence of the combination of Bernstein polynomials with a positive function τ and based on a parameter s > − 1 2. These polynomials have preserved the functions 1 and τ. First, the convergence theorem for this sequence is studied for a function f ∈ C[0, 1]. Next, the rate of convergence theorem for these polynomials is descript by using the first, second modulus of continuous and Ditzian-Totik modulus of smoothness. Also, the Quantitative Voronovskaja and Grüss-Voronovskaja are obtained. Finally, two numerical examples are given for these polynomials by chosen a test function f ∈ C[0, 1] and two functions for τ to show that the effect of the different values of s and the different chosen functions τ.

On convergence of certain nonlinear Bernstein operators

Filomat, 2016

In this article, we concern with the nonlinear Bernstein operators NBnf of the form (NBnf)(x)= n?k=0 Pn,k (x,f (k/n)), 0 ? x ? 1, n?N, acting on bounded functions on an interval [0,1], where Pn,k satisfy some suitable assumptions. As a continuation of the very recent paper of the authors [22], we estimate their pointwise convergence to a function f having derivatives of bounded (Jordan) variation on the interval [0,1]. We note that our results are strict extensions of the classical ones, namely, the results dealing with the linear Bernstein polynomials.

On approximation by a class of new Bernstein type operators

Applied Mathematics and Computation, 2008

This paper is concerned with a new type of the classical Bernstein operators where the function is evaluated at intervals ½0; 1 À 1 nþ1 . We also make extensive study simultaneous approximation by the linear combination L n ðf ; k; xÞ of these new Bernstein type operators L n ðf Þ. At the end of this paper we have given an other modification of these operators.

A New Class of Modified Bernstein Operators

Journal of Approximation Theory, 1999

The left Bernstein quasi-interpolant operator introduced by Sablonnière is a kind of modified Bernstein operator that has good stability and convergence rate properties. However, we recently found that it is not very convenient for practical applications. Fortunately, we showed in a previous paper that there exist many operators having stability and convergence rate properties similar to those of Sablonnière's operator. In this paper, we introduce another specific class of such operators generated from the operator introduced by Stancu. We present detailed results about this class, some of which can be applied to numerical quadrature. Finally, we clarify its advantages and assert that it is more natural and more convenient both theoretically and practically, than that of Sablonnière. Our paper, at the same time, provides several new results regarding Stancu's operator.