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1.
In this paper we give a complete expansion formula for Bernstein polynomials defined on ans-dimensional simplex. This expansion for a smooth functionf represents the Bernstein polynomialB
n
(f) as a combination of derivatives off plus an error term of orderO(n–s
).Communicated by Wolfgang Dahmen. 相似文献
2.
Given a real functionf C
2k
[0,1],k 1 and the corresponding Bernstein polynomials {B
n
(f)}
n
we derive an asymptotic expansion formula forB
n
(f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence thanB
n
(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.This work was partially supported by a grant from MURST 40. 相似文献
3.
This paper estimates upper and lower bounds for the approximation rates of iterated Boolean sums of multivariate Bernstein polynomials. Both direct and inverse inequalities for the approximation rate are established in terms of a certain K-functional. From these estimates, one can also determine the class of functions yielding optimal approximations to the iterated Boolean sums. 相似文献
4.
It is shown that Bernstein polynomials for a multivariate function converge to this function along with partial derivatives provided that the latter derivatives exist and are continuous. This result may be useful in some issues of stochastic calculus. 相似文献
5.
Zhang Renjiang 《分析论及其应用》1998,14(1):49-56
In this paper,an asymptotic expansion formula for approximation to a continuous function by Bernsteinpolynomials on a triangle is obtained. 相似文献
6.
Nikola Naidenov 《Analysis Mathematica》2007,33(1):55-62
There are fine extensions of the univariate Bernstein-Szeg? inequality for multivariate polynomials considered on a convex domain K. The current one estimates the gradient of the polynomial P at a point x ∈ K by constant times degree, ‖P‖ C(K) and a geometrical factor. The best constant is within [2, 2√2]. In this note we disprove the conjecture (based on some particular cases) that the best constant is 2. 相似文献
7.
8.
9.
Antonio-Jesús López-Moreno Francisco-Javier Muñoz-Delgado 《Numerical Algorithms》2005,39(1-3):237-252
In this paper we study the asymptotic expansion of sequences of multivariate Kantorovich type operators and their partial derivatives. In particular, we obtain the complete expansion for the Kantorovich Bernstein operators on the simplex and for two Kantorovich type modifications of the Bleimann, Butzer and Hahn operators that we introduce in the paper.
AMS subject classification 41A36 相似文献
10.
Note on Bernstein polynomials and Kantorovich polynomials 总被引:2,自引:0,他引:2
Wang Xiaochun 《分析论及其应用》1991,7(2):99-105
We obtain two asymptotic representations of remainder of approximation of derivable functions by Bernstein polynomials and
Kantorovich polynomials separately. 相似文献
11.
We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming
problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several
new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency
and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid
the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based
on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13
benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method
in terms of the chosen performance metrics. 相似文献
12.
Béla Nagy 《Journal of Mathematical Analysis and Applications》2005,301(2):449-456
In this paper we examine the Bernstein-Markov inequality on special compact subsets of the complex plane, namely on lemniscates. Sharp constants are obtained which involve the Green function of the complement and the density of equilibrium measure of the compact set. Using lemniscates is a useful tool because of the possibility of taking inverse images. The proof also uses so-called peaking polynomials which will be constructed. 相似文献
13.
Wu Zhengchang 《分析论及其应用》1991,7(1):81-90
In this paper, we extend the idea of linear combinations of Bernstein polynomials to multidimensional case. 相似文献
14.
15.
Yilmaz Simsek 《Applied mathematics and computation》2011,218(3):1072-1076
Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties of this generating functions are given. By applying this generating function, not only derivative of these polynomials but also recurrence relations of these polynomials are found. Interpolation function of these polynomials is also constructed by Mellin transformation. This function interpolates these polynomials at negative integers which are given explicitly. Moreover, relations between these polynomials, the Stirling numbers of the second kind and Bernoulli polynomials of higher order are given. Furthermore some remarks associated with the Bezier curves are given. 相似文献
16.
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18.
S. A. Telyakovskii 《Proceedings of the Steklov Institute of Mathematics》2008,260(1):279-286
E.V. Voronovskaya and S.N. Bernstein established an asymptotic representation for the deviation of functions from Bernstein polynomials under the condition that the function has an even-order derivative. In the present paper, a similar problem is solved in the case when the function has an odd-order derivative. In addition, analogous representations are obtained for the deviations of functions from Kantorovich polynomials. 相似文献
19.
20.
《Journal of Computational and Applied Mathematics》2006,197(2):520-533
Connection coefficients between the two-variable Bernstein and Jacobi polynomial families on the triangle are given explicitly as evaluations of two-variable Hahn polynomials. Dual two-variable Bernstein polynomials are introduced. Explicit formula in terms of two-variable Jacobi polynomials and a recurrence relation are given. 相似文献