A Bernstein-type operator approximating continuous functions on the semi-axis (original) (raw)
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.
Bernstein type operators with a better approximation for some functions
Applied Mathematics and Computation, 2013
V un f ðxÞ ¼ B n f ðu n ðxÞÞ ¼ X n k¼0 n k u n ðxÞ ð Þ k 1 À u n ðxÞ ð Þ nÀk f k n ; x 2 ½0; 1; ð2Þ where u n : ½0; 1 ! ½0; 1 are continuous functions satisfying u n ð0Þ ¼ 0 and u n ð1Þ ¼ 1. For u n ðxÞ ¼ x we get the classical Bernstein operator , and in general we have V un f ðxÞ ¼ B n f ðu n ðxÞÞ: Other operator s of the form (2) were considered in [
Approximation properties of Bernstein–Durrmeyer type operators
Applied Mathematics and Computation, 2014
This paper deals with the approximation of continuous functions by sequences of some modified Bernstein-Durrmeyer type operators that reproduce certain test functions. The orders of approximation of the new versions turn to be at least as good as the one of the genuine Bernstein-Durrmeyer operators. Moreover, by extrapolating techniques recently applied to the classical Bernstein operators, we present a one-parameter family of modified sequences of operators that reproduce certain polynomials and possess that popular genuine sequence as a limit case. Comparisons and some illustrative graphics are also presented.
On some constants in approximation by Bernstein operators
We estimate the constants sup x∈(0,1) sup f ∈C[0,1]\Π 1 |Bn(f,x)−f (x)| ω 2 f, Õ x(1−x) n and inf x∈(0,1) sup f ∈C[0,1]\Π 1 |Bn(f,x)−f (x)| ω 2 f, Õ x(1−x) n , where B n is the Bernstein operator of degree n and ω 2 is the second order modulus of continuity.
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 a New Class of Bernstein Type Operators Based on Beta Function
2020
We develop Bernstein type operators using the Beta function and study their approximation properties. By using Korovkin's theorem, we achieve the uniform convergence of sequences of these operators. We obtain the rate of convergence in terms of modulus of continuity and establish the Voronovskaja type asymptotic result for these operators. At last the graphical comparison of these newly defined operators with few of the fundamental but significant operators is discussed.
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.
On Approximation Properties of (p,q)-Bernstein Operators
European Journal of Pure and Applied Mathematics
In this study, a (p,q)-analogue of Bernstein operators is introducedand approximation properties of (p,q)-Bernstein operators areinvestigated. Some basic theorems are proved. The rate of approximationby modulus of continuity is estimated.
Better approximation results by Bernstein–Kantorovich operators
Lobachevskii Journal of Mathematics, 2017
In this paper, we give a King-type modification of the Bernstein-Kantorovich operators and study the approximation properties of these operators. We prove that the error estimation of these operators is better than the classical Bernstein-Kantorovich operators. We also give some estimations for the rate of convergence of these operators by using the modulus of continuity. Furthermore, we obtain a Voronovskaya-type asymptotic formula for these operators.
Journal of Function Spaces
In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’sK-functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-typ...
Numerical Algorithms, 2017
This paper presents a new approach to improve the order of approximation of the Bernstein operators. Three new operators of the Bernstein-type with the degree of approximations one, two, and three are obtained. Also, some theoretical results concerning the rate of convergence of the new operators are proved. Finally, some applications of the obtained operators such as approximation of functions and some new quadrature rules are introduced and the theoretical results are verified numerically.
On Approximation Properties of Generalized q-Bernstein Operators
Numerical Functional Analysis and Optimization
In this study, a (p, q)-analogue of Bernstein operators is introduced and approximation properties of (p, q)-Bernstein operators are investigated. Some basic theorems are proved. The rate of approximation by modulus of continuity is estimated.
Pointwise Approximation by Bernstein Polynomials
Bulletin of the Australian Mathematical Society, 2012
We improve the degree of pointwise approximation of continuous functions f(x) by Bernstein operators, when x is close to the endpoints of [0,1]. We apply the new estimate to establish upper and lower pointwise estimates for the test function g(x)=xlog (x)+(1−x)log (1−x). At the end we prove a general statement for pointwise approximation by Bernstein operators.