二重三角插值多项式的求和因子法求和

选取一组求和因子ρａ,β构造了二重三角插值算子Ｆｍｎ（ｆ;ｙ）,使对于任意的ｆ（ｘ,ｙ）∈Ｃ２π,２π都能在全面上一致收敛,且达到最佳收敛阶。     

Fourier级数的求和理论与方法—求和因子法求和

在 Fourier级数的线性求和中 ,通过构造求和因子 ,使得带有该求和因子的积分算子在全轴上一致地收敛到每个以 2 π为周期的连续函数 ,并对 Cj2π(0 j r)函数类的逼近均达到最佳收敛阶 ,参数 r为任意给定的奇自然数 .     

关于二元函数的三角插值逼近

本文以两组不同的节点构造了一个组合型的二元三角插值多项式算子Lmn(f;x,y),并且研究了二元连续周期函数对这个算子的收敛性及收敛阶的估计等问题。     

三角插值中的线性求和问题

本文通过选取求和因子构造出和式型三角插值多项式Ｈｎ（ｆ，ｒ，ｘ）（ｒ为奇自然数），使其在全实轴上一致地收敛到以２π为周期的连续函数ｆ（ｘ），且Ｈｎ（ｆ，ｒ，ｘ）对Ｃｎ２π（ｌ≤ｒ）连续函数类的逼近均达到最佳收敛阶．Ｈｎ（ｆ，ｒ，ｘ）的饱和阶为１／ｎ（ｒ＋１），饱和函数类为ｆ（ｒ）（ｘ）∈Ｌｉｐｍｌ．     

新二元三角插值多项式的构造

设Ω=[-πxπ,-πyπ],C(Ω)表示关于x,y均以2π为周期的连续函数空间.若f(x,y)∈C(Ω),取结点组为(xk,yl)=(2k+2n 1)π,(2l 2+m 1)πk=0,1,2,…,2n,l=0,1,2,…,2m,则我们获得一个二元三角插值多项式Cn,m(f;x,y)=M1N∑k=2n0∑l=2m0f(xk,yl).1+2∑nα=1cosα(x-xk)+2∑mβ=1cosβ(y-yl)+4∑nα=1∑mβ=1cosα(x-xk)cosβ(y-yl)其中M=2m+1,N=2n+1.为改进其收敛性,本文构造一个新的因子ρα,β,使得带有该因子ρα,β的二元三角插值多项式Ln,m(f;x,y)可以在全平面上一致地收敛到每个连续的f(x,y),且具有最佳逼近阶.     

复插值逼近

本文着重地介绍复数域上各类插值多项式(有时也提到有理函数)的收敛性与发散性问题的近代成果,适当地介绍了复插值逼近阶的估计。文章共分七节,其中§1为问题的提出;§2介绍紧集上解析函数的Lagrange插值多项式收敛的充要条件;§3介绍A(|z|≤1)上函数的Lagrange插值的收敛及发散问题;§4是一般区域上Lagrange插值的收敛问题;§5介绍调和多项式的插值;§6介绍Hermite及Hermite-Fejer插值的收敛与发散性问题;§7介绍有理函数插值的收敛性问题。     

复有理型插值一致逼近函数及其导数

本文研究在单位圆周{|z| =1}上一致逼近函数f(z)及其导数,利用Hermite插值中的基函数建立复有理型插值,并证明它们在{|z| =1}上分别一致收敛于f(z)或f′(z) ,给出了收敛速度.     

拉格朗日插值算子的Lp户收敛速度

本得到了以第二类Tchebycheff多项式的零点为插值结点组的拉格朗日插值于Lp(1相似文献   

关于级数的求和方法

高等数学关于级数的研究中，讨论了常数项级数的敛散性以及函数项级数的收敛域．但对收敛的常数项级数的求和以及在收敛域内如何求函数项级数的和函数讨论不多．级数的求和方法比较多，技巧性也比较强，下面介绍常用的有效的级数求和方法。     

Gruenwald插值算子的L1收敛速度

先给出了以第二类Ｔｃｈｅｂｙｃｈｅｆｆ多项式的零点为插值结点组的Ｇｒｕｎｗａｌｄ插值多项式于Ｌ１下的收敛速度，然后给出了一种修改的Ｇｒｕｎｗａｌｄ插值多项式及其于Ｌ１下的收敛速度。     

一个新的涉及一个多重可变上限函数和一个部分和的半离散Hilbert型不等式

应用权函数的方法,Euler-Maclaurin求和公式,Abel部分求和公式及实分析技巧,求出了一个新的涉及一个多重可变上限函数和一个部分和的半离散Hilbert型不等式.作为应用,考虑了特殊参数下不等式中最佳常数因子联系多参数的等价条件及一些特殊不等式.     

Error estimation of floating-point summation and dot product

We improve the well-known Wilkinson-type estimates for the error of standard floating-point recursive summation and dot product by up to a factor 2. The bounds are valid when computed in rounding to nearest, no higher order terms are necessary, and they are best possible. For summation there is no restriction on the number of summands. The proofs are short by using a new tool for the estimation of errors in floating-point computations which cures drawbacks of the “unit in the last place (ulp)”. The presented estimates are nice and simple, and closer to what one may expect.     

Hilbert重级数定理的一个新推广

通过引入一个适当的对数函数建立了一种新的Hilbert型不等式.利用Euler-Maclaurin求和公式对权函数进行了估计.证明了常数因子π2r＋1Er是最佳的,其中Er是Euler数.作为应用,给出了一些互相等价的不等式.     

A Dimensionally Continued Poisson Summation Formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.     

Stress intensity factors for a finite plate with an inclined crack by boundary collocation

In this paper, we combine the Muskhelishvili＇s complex variable method and boundary collocation method, and choose a set of new stress function based on the stress boundary condition of crack surface, the higher precision and less computation are reached. This method is applied to calculating the stress intensity factor for a finite plate with an inclined crack. The influence of θ （the obliquity of crack） on the stress intensity factors, as well as the number of summation terms on the stress intensity factor are studied and graphically represented.     

Rational summation ofp-adic series

The problem of rational summation for a wide class ofp-adic convergent series is considered. Here, rational summation refers to the method of obtaining the rational sum of a power series for a rational value of its variable. A formula suitable for this summation is derived. Conditions for rational summability are obtained. Rational summation is possible only for special forms of the series. It is shown that the inverse problem of rational summation is always solvable. This is illustrated by some characteristic examples. Possible rational (adelic) summation of divergent perturbative expansions in string theory, and quantum field theory, is discussed.Institute of Physics, P.O. Box 57, 11001 Belgrade, Yugoslavia. Published in Toereticheskaya i Matematicheskaya Fizika, Vol. 100, No. 3, pp. 342–353, September, 1994.     

一个逆向的Hardy-Hilbert型不等式

通过建立弱条件下的Eu ler-M aclaurin求和公式的余项估值式,精确估算权系数,给出一个逆向的H ardy-H ilbert型不等式.作为应用,考虑了它的加强式及等价形式.     

Accelerating Indefinite Summation: Simple Classes of Summands

We present the history of indefinite summation starting with classics (Newton, Montmort, Taylor, Stirling, Euler, Boole, Jordan) followed by modern classics (Abramov, Gosper, Karr) to the current implementation in computer algebra system Maple. Along with historical presentation we describe several “acceleration techniques” of algorithms for indefinite summation which offer not only theoretical but also practical improvement in running time. Implementations of these algorithms in Maple are compared to standard Maple summation tools.     

Summation of Unordered Arrays

An approach to the summation of unordered number and matrix arrays based on ordering them by absolute value (greedy summation) is proposed. Theorems on products of greedy sums are proved. A relationship between the theory of greedy summation and the theory of generalized Dirichlet series is revealed. The notion of asymptotic Dirichlet series is considered.