多元λ-矩阵行列式的FFT展开法

我们知道,直接展开一个λ-矩阵的行列式,其工作量是很大的。对于多元多项式矩阵(即每个元素为多元多项式的矩阵)的行列式展开,工作量则更为惊人。本文利用多维FFT得到了求多元多项式矩阵行列式的一个简单快速的计算方法,并估计了计算复杂性的上界。     

Z／mZ上的多元置换多项式

本文研究了一类典型的模ｐ的多元奇异多项式，得到了它们是模Ｐｌ（ｌ＞１）的置换多项式的充要条件并给出了一个是模ｐ２的置换多项式但不是模ｐ３的置换多项式的多元多项式例子，从而说明模ｐｌ（ｌ＞１）的多元置换多项式不能（象一元那样）简化到模ｐ上．     

二元整系数多项式因式分解的一种理论和算法

将二元多项式看成系数为一元多项式的一元多项式来进行分解，本文建立了二元整系数多项式因式分解的一种理论，提出了一个完整的分解二元整系数多项式的新算法．这个算法能自然地推广到多元整系数多项式的分解中去．     

一种广义周期Besov类的多元周期多项式样条逼近

本文证明多元多项式周期样条空间是某些多元周期光滑函数类的关于Kolmogorov n-宽度的弱渐近极子空间．给出了广义周期Besov类的一种推广，得到了空间元素的一种表示定理，不仅给出了一种多元周期多项式样条算子．而且证明了所得的结果．     

GF（3）上多元多项式的化简

本文通过极性矩阵的递归表示，对ＧＦ（３）上多元多项式环进行了讨论，提出将变量经过线性变换，使多元多项式化简为乘积项数最少的新方法．该方法不需要进行矩阵运算，简便易行，并减少了计算复杂性，其结果改进了［１，２］的工作．     

多元Lagrange插值与Cayley-Bacharach定理

1 引言多元Lagrange插值一直是计算数学中一个重要的研究课题．为了解决一些实际科学计算问题(如多元函数的计算,曲面的外形设计和有限元格式的建立等),有关多元多项式插值的理论与方法的研究在近二、三十年中迅速发展起来．在研究多元多项式插值时, 一个首先必须解决的问题就是多元插值的适定性问题．目前,国内外对这一问题的研究大     

二元整系数多项式因式分解的一种理论和算法

我们发现可以把二元多项式盾成系数为一元多项式的一元多项式来进行分解，据此，本文建立了二元整系数多项式因式分解的一种理论，提出了一个完整的分解二元整系数多项式的算法。这个算法还能很自然地推广成分解多元整系数多项式的算法。     

第十二讲 对称多项式

关于对称多项式的研究是近代数学的主要分支群论产生的直接原因。中学生了解对称多项式的性质,对于今后理解群论的基本概念和思想,无疑是有好处的。中学生数学竞赛试题常常含有对称式方面的题目。定义1 含有n个变元的多项式谓之n元多项式,记为f(x_1,x_2,…x_n)。例如.x~2-y~2是二元二次多项式,3x_1~2x_2~2+2x_1x_2~2x_3+x_3~3是三元四次多项式,x~3+y~3+z~3-3xyz是三元三次多项式。对一元多项式我们常采用降幂(或升幂)排列。对于多元多项式我们经常采用字典排列法,即对于n元多项式的两个单项式ax_1~(k_2)x2~(k_2) …x_n~(k_n)和     

高阶多元Euler多项式和高阶多元Bernoulli多项式

本文给出了高阶多元Ｅｕｌｅｒ数和多项式与高阶多元Ｂｅｒｎｏｕｌｉ数和多项式的定义，讨论了它们的一些重要性质，得到了高阶多元Ｅｕｌｅｒ多项式（数）和高阶多元Ｂｅｒｎｏｕｌｉ多项式（数）的关系式·     

高阶多元Euler多项多和高阶多元Bernoulli多项式

本给出了高阶多元Ｅｕｌｅｒ数和多项式与同阶多元Ｂｅｒｎｏｕｌｌｉ数和多项式的定义，讨论了它们的一些重要性质，得到了高阶多元Ｅｕｌｅｒ多项式（数）和高阶多元Ｂｅｒｎｏｕｌｌｉ多项式（数）的关系式。     

Reconstruction of Sparse Polynomials via Quasi-Orthogonal Matching Pursuit Method

In this paper, we propose a Quasi-Orthogonal Matching Pursuit (QOMP) algorithm for constructing a sparse approximation of functions in terms of expansion by orthonormal polynomials. For the two kinds of sampled data, data with noises and without noises, we apply the mutual coherence of measurement matrix to establish the convergence of the QOMP algorithm which can reconstruct $s$-sparse Legendre polynomials, Chebyshev polynomials and trigonometric polynomials in $s$ step iterations. The results are also extended to general bounded orthogonal system including tensor product of these three univariate orthogonal polynomials. Finally, numerical experiments will be presented to verify the effectiveness of the QOMP method.     

On the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials

The aim of this paper is to investigate the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials. Starting from the Lagrange interpolation polynomials, we shall construct a wide class of discrete processes which are uniformly convergent in a suitable weighted space of continuous functions. Error estimates for the approximation will also be considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

单纯形上的Stancu多项式与最佳多项式逼近

作为Bernstein多项式的推广,本文定义单纯形上的多元Stancu多项式.以最佳多项式逼近为度量,建立Stancu多项式对连续函数的逼近定理与逼近阶估计,给出Stancu多项式的一个逼近逆定理,从而用最佳多项式逼近刻划Stancu多项式的逼近特征.     

On the Relation between Piecewise Polynomial and Rational Approximation in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Bold">R</Emphasis><Superscript>2</Superscript>)

In the univariate case there are certain equivalences between the nonlinear approximation methods that use piecewise polynomials and those that use rational functions. It is known that for certain parameters the respective approximation spaces are identical and may be described as Besov spaces. The characterization of the approximation spaces of the multivariate nonlinear approximation by piecewise polynomials and by rational functions is not known. In this work we compare between the two methods in the bivariate case. We show some relations between the approximation spaces of piecewise polynomials defined on n triangles and those of bivariate rational functions of total degree n which are described by n parameters. Thus we compare two classes of approximants with the same number Cn of parameters. We consider this the proper comparison between the two methods.     

On the Convergence of Bernstein-Stancu Polynomials in the Variation Seminorm

This article studies the variation detracting property and rate of approximation of the Bernstein-Stancu polynomials in the space of functions of bounded variation with respect to the variation seminorm. Moreover, we will present Voronovskaya-type theorems for Bernstein-Stancu polynomials B_{n, α, β}f and for the first derivative of these polynomials. Finally we include some graphical examples.     

Operator adapted spectral element methods I: harmonic and generalized harmonic polynomials

Summary. The paper presents results on the approximation of functions which solve an elliptic differential equation by operator adapted systems of functions. Compared with standard polynomials, these operator adapted systems have superior local approximation properties. First, the case of Laplace's equation and harmonic polynomials as operator adapted functions is analyzed and rates of convergence in a Sobolev space setting are given for the approximation with harmonic polynomials. Special attention is paid to the approximation of singular functions that arise typically in corners. These results for harmonic polynomials are extended to general elliptic equations with analytic coefficients by means of the theory of Bergman and Vekua; the approximation results for Laplace's equation hold true verbatim, if harmonic polynomials are replaced with generalized harmonic polynomials. The Partition of Unity Method is used in a numerical example to construct an operator adapted spectral method for Laplace's equation that is based on approximating with harmonic polynomials locally. Received May 26, 1997 / Revised version received September 21, 1998 / Published online September 7, 1999     

On the best approximation of vector valued functions by polynomials with coefficients in vector spaces

In this paper, we prove some basic results concerning the best approximation of vector-valued functions by algebraic and trigonometric polynomials with coefficients in normed spaces, called generalized polynomials. Thus we obtain direct and inverse theorems for the best approximation by generalized polynomials and results concerning the existence (and uniqueness) of best approximation generalized polynomials. This paper was written during the 2005 Spring Semester when the second author (S.G. Gal) was a Visiting Professor at the Department of Mathematical Sciences, The University of Memphis, TN, USA. Mathematics Subject Classification (2000) 41A65, 41A17, 41A27     

The Genuine Bernstein-Durrmeyer Operator on a Simplex

In 1967 Durrmeyer introduced a modification of the Bernstein polynomials as a selfadjoint polynomial operator on L²[0,1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer’s modification, and identified these operators as de la Vallée-Poussin means with respect to the associated Jacobi polynomial expansion. Nevertheless, all these modifications lack one important property of the Bernstein polynomials, namely the preservation of linear functions. To overcome this drawback a Bernstein-Durrmeyer operator with respect to a singular Jacobi weight will be introduced and investigated. For this purpose an orthogonal series expansion in terms generalized Jacobi polynomials and its de la Vallée-Poussin means will be considered. These Bernstein-Durrmeyer polynomials with respect to the singular weight combine all the nice properties of Bernstein-Durrmeyer polynomials with the preservation of linear functions, and are closely tied to classical Bernstein polynomials. Focusing not on the approximation behavior of the operators but on shape preserving properties, these operators we will prove them to converge monotonically decreasing, if and only if the underlying function is subharmonic with respect to the elliptic differential operator associated to the Bernstein as well as to these Bernstein-Durrmeyer polynomials. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved by these Bernstein-Durrmeyer polynomials. Finally, pointwise and global saturation results will be derived in a very elementary way.     

Best one-sided L1 approximation to the Heaviside and sign functions

We find the polynomials of the best one-sided approximation to the Heaviside and sign functions. The polynomials are obtained by Hermite interpolation at the zeros of some Jacobi polynomials. Also we give an estimate of the error of approximation and characterize the extremal points of the convex set of the best approximants.     

Approximation by Entire Functions on a Countable Union of Segments on the Real Axis: 3. Further Generalization

In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Hölder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.