Some new properties of Generalized 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. .

Honam Mathematical Journal, 2011

We prove two identities for multivariate Bernstein polynomials on simplex, which are considered on a pointwise. In this paper, we study good approximations of Bernstein polynomials for every continuous functions on simplex and the higher dimensional q-analogues of Bernstein polynomials on simplex.

Journal of Mathematical Analysis and Applications, 2007

Explicit formulae, in terms of Bernstein-Bézier coefficients, of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on an interval are derived in this paper. Examples are given and discussed to illustrate the general theory. Finally, this consideration is related to the study of refinement masks of spline function vectors.

2011

In this paper, derivatives of the product of Bernstein polynomials of the same and different degrees are obtained. Also a recurrence formula for those polynomials together with some new properties are given.

Computers & Mathematics with Applications, 1995

This paper discusses the criteria of convexity, monotonicity, and positivity of Bernstein-B~zier polynomials over simplexes.

2004

We introduce polynomials B n i (x; ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B n i (x; ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented.

2008

In this article we want to determinate a recursive formula for Bernstein polynomials associated to the functions ep(x) = xp, p ∈ N, and an expresion for the central moments of the Bernstein polinomyals. 2000 Mathematics Subject Classification. 41A10; 41A63.

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.

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 .

Mathematical Methods in the Applied Sciences, 2014

The purpose of this paper is to define a new class polynomials. Special cases of these polynomials give many famous family of the Bernstein type polynomials and beta polynomials. We also construct generating functions for these polynomials. We investigate some fundamental properties of these functions and polynomials. Using functional equations and generating functions, we derive various identities related to theses polynomials. We also construct interpolation function that interpolates these polynomials at negative integers. Finally, we give a matrix representations of these polynomials. Copyright

The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of two variables. By using these type polynomials, we derive recurrence formulas and some new interesting identities related to the second kind Stirling numbers and generalized Bernoulli polynomials. Moreover, we give the generating function and interpolation function of these modified q-Bernstein polynomials of two variables and also give the derivatives of these polynomials and their generating function

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.

Applied Mathematics and Computation, 2010

Bernstein-Stancu type polynomials Modulus of continuity Uniform approximation Rate of convergence Positive linear operators Korovkin's theorem a b s t r a c t A new generalization of Bernstein-Stancu type polynomials for one and two variables are constructed and the theorems on convergence and the degree of convergence are established. In addition some numerical examples, corresponding to obtaining results are given.

Journal of Approximation Theory, 2006

The evaluation of multivariate polynomials of n variables in Bernstein-Bézier form is considered. A forward error analysis for the corresponding de Casteljau algorithm and the VS algorithm is performed. We also include algorithms that simultaneously evaluate the polynomial and provide "a posteriori" error bounds, without increasing significantly the computational cost. The sharpness of our running error bounds is shown in the case of trivariate polynomials.

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 .