The Degree Of Polynomial Approximation And Interpolation Of Analytic Functions

The rate of best polynomial approximation of an analytic function on a compact Faber set K is characterized in terms of the rate of growth of its Faber coefficients and compared with the rate of approximation by the partial sums of the Faber series. Also the convergence of sequences of interpolating polynomials constructed for various systems of nodes is studied by considering the growth of the interpolated function. Under appropriate assumptions on K the approximation by interpolating polynomials can be incorporated in the characterization theorem. Emphasis is laid on high precision in describing the rate of approximation and on admitting a large class of functions.     

奇次Legendre多项式零点上的(0,1,3)插值的正则性

主要讨论了奇次Legendre多项式零点上的(0,1,3)插值的正则性,得出它是奇异的,但当对插值的条件作适当的改变后也得到正则的结论.     

Nonexistence of Isochronous Centers in Planar Polynomial Hamiltonian Systems of Degree Four

In this paper we study centers of planar polynomial Hamiltonian systems and we are interested in the isochronous ones. We prove that every center of a polynomial Hamiltonian system of degree four (that is, with its homogeneous part of degree four not identically zero) is nonisochronous. The proof uses the geometric properties of the period annulus and it requires the study of the Hamiltonian systems associated to a Hamiltonian function of the form H(x, y)=A(x)+B(x) y+C(x) y²+D(x) y³.     

多项式插值与Rolle定理

本文利用多项式插值 ,对微分学和积分学中的一些典型题目 ,给出了统一的处理方法 .这种方法易于模仿 ,有一定的适用性 .     

新组合型的三角插值多项式

将被插函数进行组合平均,构造一个新组合型的三角插值多项式Cn(f;t,x),使得它在全轴上一致收敛到每个以2π为周期的连续函数,且对Cj2π连续函数类的逼近阶达到最佳,这里0jt,t为任给的奇自然数.     

一类含三角形图的伴随多项式的根

We denote h(G,x) as the adjoint polynomial of graph G. In [5], Ma obtained the interpolation properties of the roots of adjoint polynomial of graphs containing triangles. By the properties, we prove the non-zero root of adjoint polynomial of Dn and Fn are single multiple.     

Polynomial Interpolation of the Discrete Logarithm

Recently, Coppersmith and Shparlinski proved several results on the interpolation of the discrete logarithm in the finite prime field by polynomials modulo p and modulo p-1, respectively. In this paper most of these results are extended to arbitrary .     

关于Bernstein的一个插值多项式

本文证得(?)｜Ｐ_ｎ（ｆ，ｘ）｜＝＋∞，因此Ｐ_ｎ（ｆ，ｘ）不能对一切ｆ（ｘ）Ｃ_{［－１，１］}在［－１，１］上一致收敛于ｆ（ｘ）     

Bivariate Polynomial Interpolation over Nonrectangular Meshes

In this paper, by means of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.     

Length Four Polynomial Automorphisms

We study the structure of length four polynomial automorphisms of R[X, Y] when R is a unique factorization domain. The results from this study are used to prove that, if SL _m (R[X ₁, X ₂,…, X _n ]) = E _m (R[X ₁, X ₂,…, X _n ]) for all n, m ≥ 0, then all length four polynomial automorphisms of R[X, Y] that are commutators are stably tame.     

关于一类修正的三角插值多项式

本文构造出一个以｛θ_ｋ＝ｋ/(ｎ＋１)π｝^ｎ_ｋ＝１为插值节点的ｆ（θ）∈Ｃ_２π且为奇函数的修正的三角插值多项式Ｗ_n（ｆ;ｒ,θ）（ｒ为自然数）．Ｗ_n（ｆ;ｒ,θ）对每个以２π为周期的奇连续函数都能在全实轴上一致地收敛到ｆ（θ）;若ｆ（θ）∈Ｃ^ｊ_２π（０≤ｊ≤ｒ－１）且是奇的,Ｗ_n（ｆ;ｒ,θ）对其收敛阶均达到最     

Lines Tangent to Four Triangles in Three-Dimensional Space

We investigate the lines tangent to four triangles in R³. By a construction, there can be as many as 62 tangents. We show that there are at most 162 connected components of tangents, and at most 156 if the triangles are disjoint. In addition, if the triangles are in (algebraic) general position, then the number of tangents is finite and it is always even.     

以线性代数观点看常用多项式插值方法

数学各分支之间存在很多共同的思想和方法.本文结合线性代数中关于线性空间的相关理论及观点来看数值分析中常用的多项式插值方法.一方面借助线性空间的基与坐标理论将常用多项式插值方法统一起来,并借助过渡矩阵给出了不同插值方法之间的通用转换公式.另一方面还可以通过构造特殊基函数组来产生新的多项式插值方法.     

The General Problem of Polynomial Spline Interpolation

We study the general problem of interpolation by polynomial splines and consider the construction of such splines using the coefficients of expansion of a certain derivative in B-splines. We analyze the properties of the obtained systems of equations and estimate the interpolation error.     

一类插值多项式的导数逼近(英文)

本文给出了以雅可比多项式的零点作为插值节点的一类插值多项式 Bn( f ;x)的导数逼近具有一阶连续导数的函数的收敛阶 .并且指出 limn→∞ Bn′( f;-1 )≠f′( -1 ) .     

Degree of Simultaneous Coconvex Polynomial Approximation

$ {\rm Let}\ f\ \epsilon\ {C^{1}}[-1,1] $ change its convexity finitely many times in the interval, say s times, at ${\rm at}\ {Y_{s}}\:\ -1\ <\ y_{s}\ <\ \dots\ < y_{1}\ < 1 $ . We estimate the degree of simultaneous approximation of ? and its derivative by polynomials of degree n, which change convexity exactly at the points Y _s, and their derivatives. We show that provided n is sufficiently large, depending on the location of the points Y _s, the rate of approximation can be estimated by C(s)/n times the second Ditzian-Totik modulus of smoothness of ?′. This should be compared to a recent paper by the authors together with I. A. Shevchuk where ? is merely assumed to be continuous and estimates of coconvex approximation are given by means of the third Ditzian-Totik modulus of smoothness. However, no simultaneous approximation is given there.     

Polynomial functions and the Riesz Interpolation Property

Building on a proof by D. Handelman of a generalisation of an example due to L. Fuchs, we show that the space of real-valued polynomials on a non-empty set X$X$ of reals has the Riesz Interpolation Property if and only if X$X$ is bounded.     

Hermite插值多项式在不同基下的显式表示

本文利用对偶基的概念,导出了 Ｈｅｒｍ ｉｔｅ 插值多项式在不同基下的显式表示,这给人们对 Ｈｅｒｍ ｉｔ插值多项式在不同基下从一种表示转换到另一种表示带来极大的方便     

多项式插值算法的舍入误差分析

本文定义了多项式插值算子的条件数和多项式插值算法的数值稳定性等概念.主要研究结果是:若n和Y_max不太大,当结点等距分布时,Lagrange插值和Newton插值算法都是数值稳定的.但是不论结点如何分布,上述两法的外推计算可能是数值不稳定的.文中数值例子验证了这些理论结果.     

ASYMPTOTIC REPRESENTATION FOR REMAINDER OF QUASI-HERMITE-FEJER INTERPOLATION POLYNOMIAL

Let Q_(2n+1)(f,x)be the quasi-Hermite-Fejer interpolation polynomial of functionf(x)∈C_[-1,1]based on the zeros of the Chebyshev polynomial of the second kind U_n(x)=sin((n+l)arccosx)/sin(arc cosx). In this paper, the uniform asymptotic representation for thequantity| Q_(2n+l)(f, x) -f(x) |is given. A similar result for the Hermite-Fejer interpolationpolynomial based on the zeros of the Chebyshev polynomial of the first kind is alsoestablished.