三角域上分片二次函数Bernstein多项式的退化性

设Ｓ_２（Ｔ）为三角域Ｔ的二阶剖分，本文给出在Ｓ_２（Ｔ）下分片二次函数f(P)∈Ｃ¹（Ｔ）的Ｂｅｒｎｓｔｃｉｎ多项式的退化性及递推公式。这里的条件Ｓ_２（Ｔ）及Ｃ¹（Ｔ）类都是重要的。我们举例说明更一般情况下分片二次函数Ｂｅｒｎｓｔｃｉｎ多项式的复杂性。     

Bernstein型多项式逼近的逆定理

对于Bernstein型多项式，利用强Voronovskaja型展开，证明该多项式逼近连续函数强型逆定理，从而用Ditzian-Totik模刻画该多项式逼近阶的特征，得到了等价刻画定理．     

Bernstein多项式的升阶定理

本文给出了Ｂｅｒｎｓｔｅｉｎ多项式升阶定理及其凸性、正性的说明，同时讨论了多项式凸与网凸的关系。     

二元Bernstein多项式的Lipschitz常数

<正> 我们说f∈Lip_(Aμ)是指 |f(x_1,y_1)-f(x_2,y_2)|≤A|x_1-x_2|~μ+|y_1-y_2|~μ)对任何(x_1,y_1),(x_2,y_2)∈T成立。这里0<μ≤1,A是与f和μ有关的Lipschitz常数。     

关于Bernstein多项式的一个注记

本给出了Z.Ditzian定理的一个简短证明。     

三角形上Bernstein多项式的导数逼近

本文研究了三角域上的非乘积型Ｂｅｒｎｓｔｅｉｎ多项式的导数逼近函数时的收敛阶估计及其迭代极限．     

关于Bernstein型多项式导数的特征

利用高阶光滑模研究Bernstein型多项式的高阶导数问题，用函数的光滑性刻画Bernstein型多项式的高阶导数的特征，得到了一个等价定理。     

单纯形上Bernstein多项式凸性定理的逆定理的一个简短证明

本文给出了张景中、常庚哲、杨路所证明的《高维单纯形上Bernstein多项式凸性定理的逆定理》的一个简短证明.     

单纯形上Bernstein多项式的一些性质

设Ｂ_m（ｆ，·）为函数ｆ在ｄ维单纯形σ上的ｎ阶Ｂｅｒｎｓｔｅｉｎ多项式，本文对ｆ∈Ｃ^ｒ（σ）及ｆ∈Ｃ^r+2（σ）给出了ｆ的各阶编导数用Ｂ_n（ｆ，·）相应偏导数逼近的误差估计．同时也考虑了整系数Ｂｅｒｎｓｔｅｉｎ多项式的Ｌ_p模估计     

任意三角形区域上Bernstein多项式的Lipschitz常数

<正> 1.设f∈C[0,1],B_n(f;x)为f的Bernstein多项式,即     

Bernstein Operators for Exponential Polynomials

Let L be a linear differential operator with constant coefficients of order n and complex eigenvalues λ ₀,…,λ _n . Assume that the set U _n of all solutions of the equation Lf=0 is closed under complex conjugation. If the length of the interval [a,b] is smaller than π/M _n , where M _n :=max {|Im λ _j |:j=0,…,n}, then there exists a basis p _n,k , k=0,…,n, of the space U _n with the property that each p _n,k has a zero of order k at a and a zero of order n−k at b, and each p _n,k is positive on the open interval (a,b). Under the additional assumption that λ ₀ and λ ₁ are real and distinct, our first main result states that there exist points a=t ₀<t ₁<⋅⋅⋅<t _n =b and positive numbers α ₀,…,α _n , such that the operator

有限域上由一组对角多项式确定的簇中的点

设F_q为有限域,f_i(x)=a_(i1)x_1~(d_(i1))+…+a_(in)x_n~(d_(in))+c_i(i=1,…,m)为F_q上一组对角多项式,用N(V)表示由f_i(i=1,…,m)确定的簇中的F_q．有理点的个数．通过应用Adolphson和Sperber所引进的牛顿多面体方法,证明了ord_qN(V)≥[1/d_1+…+1/d_n]-m,其中d_i=max{d_(1i),…,d_(mi)}．该结果在许多情形下可以改进Ax- Katz定理,并推广了Wan在m=1时得到的一个定理,而且我们对Wan的定理给出了一个不同的证明．     

The best Lipschitz constants of Bernstein polynomials and Bezier nets over a given triangle

This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i.e. f(P)∃Lip_Aα, then the corresponding Bernstein Bezier net f_n∃Lip _Asec ^αφα, here φ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net f_n∃Lip _Bα, then its elevation Bezier net Ef_n∃Lip _Bα; and (3) If f(P)∃Lip _Aα, then the corresponding Bernstein polynomials B_n(f;P)∃Lip _Asec ^αφα, and the constant Asec^αφ is best in some sense. Supported by NSF and SF of National Educational Committee     

单纯型上广义Bernstein算子

该文引进并研究定义在n维单纯型上的广义Bernstein算子.首先,证明该算子具有对称性和保持Lipshcitz性质.其次,借助多元Ditzian-Totik连续模,得到该算子逼近连续函数的一个强型正向估计和一个弱型逆向不等式.最后,给出参数sn满足不同条件的若干Voronovskaja型展开式.该文所获得的结果包含了经典的Bernstein算子的相应结果.     

Direct and Inverse Estimates for Bernstein Polynomials

Direct estimates for the Bernstein operator are presented by the Ditzian—Totik modulus of smoothness , whereby the step-weight φ is a function such that φ ² is concave. The inverse direction will be established for those step-weights φ for which φ ² and , are concave functions. This combines the classical estimate (φ=1 ) and the estimate developed by Ditzian and Totik ( ). In particular, the cases , λ∈[0,1] , are included. August 2, 1996. Date revised: March 28, 1997.     

On the Bernstein Constants of Polynomial Approximation

Assume is not an integer. In papers published in 1913 and 1938, S.~N.~Bernstein established the limit

Evaluating Lipschitz Constants for Functions Given by Algorithms

This paper describes an efficient method (O(n)) to evaluate the Lipschitz constant for functions described in some algorithmic language. Considering arithmetical operations as the basis of the algorithmic language and supported by control structures, the rules to evaluate such Lipschitz constants are presented and their correctness is proved. An extension of the method to evaluate Lipschitz constants over interval domains is also presented. Examples are presented, but the effectiveness of the method is doubtful when compared to other approaches, and effective enhancements based on slope evaluations are also explored.     

Pointwise Approximation Theorems for Combinations and Derivatives of Bernstein Polynomials

We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness wФ^r（f, t） where Ф is an admissible step-weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.