Asymptotic expansion and extrapolation for Bernstein polynomials with applications

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.     

Intertwining unisolvent arrays for multivariate Lagrange interpolation

Generalizing a classical idea of Biermann, we study a way of constructing a unisolvent array for Lagrange interpolation in C^n+m out of two suitably ordered unisolvent arrays respectively in Cⁿ and C^m. For this new array, important objects of Lagrange interpolation theory (fundamental Lagrange polynomials, Newton polynomials, divided difference operator, vandermondian, etc.) are computed. AMS subject classification 41A05, 41A63     

竖线型结点组上的插值及向高维情形的推广

本文讨论了Ｒ＾２中竖线型结点组插值的适定性，得到了相应的插值多项式，并将这些结果推广到Ｒ＾ｓ（ｓ〉２）的情形。     

Asymptotic behavior of the Eckhoff method for convergence acceleration of trigonometric interpolation

Convergence acceleration of the classical trigonometric interpolation by the Eckhoff method is considered, where the exact values of the jumps are approximated by solution of a system of linear equations. The accuracy of the jump approximation is explored and the corresponding asymptotic error of interpolation is derived. Numerical results validate theoretical estimates.     

一类多元全次数 Hermite 插值问题研究

In this paper we study a class of multivariate Hermite interpolation problem on 2~d nodes with dimension d ≥ 2 which can be seen as a generalization of two classical Hermite interpolation problems of d = 2. Two combinatorial identities are firstly given and then the regularity of the proposed interpolation problem is proved.     

一种修正的拟Grünwald插值在Orlicz空间内的逼近度

修正了以第二类Chebyshev多项式的零点为插值结点组的拟Grünwald插值多项式,使之转化为积分形式,并利用不等式技巧和Hardy-Littlewood极大函数的方法,研究了此积分型拟Grünwald插值算子在带权Orlicz空间内的逼近问题,得出了意义相对广泛的逼近度估计的结果.     

Asymptotic formulae of Mehler–Heine-type for certain classical polyorthogonal polynomials

The purpose of this article is to give some asymptotic formulae of polyorthogonal polynomials with respect to some classical measures. The formulae are analogous to the Mehler–Heine formulae for Jacobi and Laguerre polynomials.     

有关调和数的更多渐近展开公式

基于指数型完全Bell多项式,建立了一个一般调和数渐近展开式,并给出展开式中系数的相应递推关系.由生成函数方法进一步推导出这些系数的具体表达式.另外,我们建立了两个在对数项里只含有奇数或偶数次幂项的lacunary调和数渐近展开式,     

The Asymptotic Behavior of Certain Birth Processes

We describe a connection between discrete birth process and a certain family of multivariate interpolation polynomials. This enables us to compute all asymptotic moments of the birth process, generalizing previously known results for the mean and variance. Received July 15, 2004     

Asymptotic behaviour of Verblunsky coefficients

Let , ζh≠ζk, h≠k and |ζj|=1, j=1,…,m, and consider the polynomials orthogonal with respect to , , where μ is a finite positive Borel measure on the unit circle with infinite points in its support, such that the reciprocal of its Szeg? function has an analytic extension beyond |z|<1. In this paper we deduce the asymptotic behaviour of their Verblunsky coefficients. By means of this result, an asymptotic representation for these polynomials inside the unit circle is also obtained.     

Asymptotic analysis of the Krawtchouk polynomials by the WKB method

We analyze the Krawtchouk polynomials K _n (x,N,p,q) asymptotically. We use singular perturbation methods to analyze them for N→∞, with appropriate scalings of the two variables x and n. In particular, the WKB method and asymptotic matching are used. We obtain asymptotic approximations valid in the whole domain [0,N]×[0,N], involving some special functions. We give numerical examples showing the accuracy of our formulas.      

Multivariate Vandermonde Determinants and General Birkhoff Interpolation

In this paper, we consider the Straight Line Type Node Configuration C (SLTNCC) in multivariate polynomial interpolation as the result of different kinds of transformations of lines (such as parallel translations, rotations). Corresponding to these transformations we define different kinds of interpolation problems for the SLTNCC. The expression of the confluent multivariate Vandermonde determinant of the coefficient matrix for each of these interpolation problems is obtained, and from this expression we conclude the related interpolation problem is unisolvent. Also, we give a kind of generalization of the SLTNCC in Section 5. As well, we obtain an expression of the interpolating polynomial for a kind of interpolation problem discussed in this paper.     

On the construction of interpolation mesh surfaces

A method for constructing two-dimensional interpolation mesh functions is proposed that is more flexible than the classical cubic spline method because it makes it possible to construct interpolation surfaces that fit the given function at specified points by varying certain parameters. The method is relatively simple and is well suited for practical implementation.     

Sheffer多项式的渐近展开

本文利用组合分析中的循环指示表示方法,找到了Ｓｈｅｆｆｅｒ型多项式的渐近展开公式及余项估计,文末讨论了所得渐近公式的运用范围,     

Markov-type Inequalities for Surface Gradients of Multivariate Polynomials

Let K ^d be a compact set with a smooth boundary and consider a polynomial p of total degree n such that p_C(K)1. Then we show that D_Tp(x)=o(n²) for any x Bd K and T a tangential direction at x. Moreover, the o(n²) term is given in terms of the modulus of smoothness of Bd K.     

Degree Conditions for Spanning Brooms

A broom is a tree obtained by subdividing one edge of the star an arbitrary number of times. In (E. Flandrin, T. Kaiser, R. Ku?el, H. Li and Z. Ryjá?ek, Neighborhood Unions and Extremal Spanning Trees, Discrete Math 308 (2008), 2343–2350) Flandrin et al. posed the problem of determining degree conditions that ensure a connected graph G contains a spanning tree that is a broom. In this article, we give one solution to this problem by demonstrating that if G is a connected graph of order with , then G contains a spanning broom. This result is best possible.     

A Markov-Type Inequality for Multivariate Polynomials on a Convex Body

A Markov-type inequality for the k-homogeneous part of a multivariate polynomial on a convex centrally symmetric body is given and an extremal polynomial is found. This generalizes and extends some estimates for univariate and multivariate polynomials obtained by Markov, Bernstein, Visser, Reimer, and Rack.     

Local Polynomial Regression: Optimal Kernels and Asymptotic Minimax Efficiency

We consider local polynomial fitting for estimating a regression function and its derivatives nonparametrically. This method possesses many nice features, among which automatic adaptation to the boundary and adaptation to various designs. A first contribution of this paper is the derivation of an optimal kernel for local polynomial regression, revealing that there is a universal optimal weighting scheme. Fan (1993, Ann. Statist., 21, 196-216) showed that the univariate local linear regression estimator is the best linear smoother, meaning that it attains the asymptotic linear minimax risk. Moreover, this smoother has high minimax risk. We show that this property also holds for the multivariate local linear regression estimator. In the univariate case we investigate minimax efficiency of local polynomial regression estimators, and find that the asymptotic minimax efficiency for commonly-used orders of fit is 100% among the class of all linear smoothers. Further, we quantify the loss in efficiency when going beyond this class.     

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

We analyze the Charlier polynomials C _n (χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.