An Improved Method for the Solution of the Heat Equation in Chebyshev Series

A method is given for the numerical solution of the heat equationusing Chebyshev series. A feature of the method is that thematrices which have to be inverted (apart possibly from a few22 matrices) are all triangular. The method shows a considerableimprovement in accuracy over previous Chebyshev methods, particularlyas far as the approximation to the fundamental eigenvalue isconcerned. The extension of the method to equations of the form is also described.     

Error Estimate of the Fourier Collocation Method for the Benjamin-Ono Equation

In this paper, the Fourier collocation method for solving the generalized Benjamin-Ono equation with periodic boundary conditions is analyzed. Stability of the semi-discrete scheme is proved and error estimate in H1/2-norm is obtained.     

阶梯柱屈曲的改进Fourier级数分析

该文对阶梯柱的弹性屈曲问题进行了研究.首先基于改进Fourier级数法采用局部坐标逐段建立阶梯柱的位移函数表达式,然后由带约束的势能变分原理得到含屈曲荷载的线性方程组,利用线性方程组有非零解的条件把问题转化为矩阵特征值问题得到临界载荷,最后讨论方法中的参数取值,并把结果与已有文献和有限元的结果比较,从而验证方法的精度.所提模型在阶梯柱的两端和变截面处引入横向弹簧和旋转弹簧,通过改变弹簧的刚度值模拟不同的边界.所提方法在工程设计中能比较精确地确定各种弹性边界条件下阶梯柱的临界载荷.     

On Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group C ^*-algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.     

两种基本函数项级数系数公式注记

分析部分高等数学教材在推导幂级数和傅里叶级数的系数公式时存在的不足,并从函数逼近论的角度,就此问题提出改进思想和方法.     

On Fourier Series on the Torus and Fourier Transforms

Mathematical Notes - The question of the representability of a continuous function on $$mathbb R^d$$ in the form of the Fourier integral of a finite Borel complex-valued measure on $$mathbb R^d$$...     

An Edge Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Let f be a function periodic in its two variables x, y with period 2 and having at the origin an edge point jump discontinuity (its graph may be roughly described by a cliff with a smooth but possibly inclined rim). The Fourier series of f exhibits a Gibbs phenomenon as in the one-dimensional case but with an affine horizontal deformation depending on the compass direction of the rim at the origin.     

An Edge Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Let f be a function periodic in its two variables x, y with period 2 and having at the origin an edge point jump discontinuity (its graph may be roughly described by a cliff with a smooth but possibly inclined rim). The Fourier series of f exhibits a Gibbs phenomenon as in the one-dimensional case but with an affine horizontal deformation depending on the compass direction of the rim at the origin.Received February 7, 2002; in revised form May 24, 2002 Published online April 4, 2003     

Sets of Divergence for Fourier Series

There exists a G set such that no Fourier series can divergeon it and converge off it.     

RAS法真实误差的实证研究

RAS法为更新投入产出表的主要非调查方法之一,作为非调查方法,更新误差的大小决定了方法的应用价值,但传统的误差估计方法由于在设计上的缺陷,其估计结果在本质上是RAS法误差的下限.本文利用1992、1997、2002年中国投入产出表及1997、2002年常规统计数据首次对实际应用中RAS法的真实误差进行了实证研究.结果表明:传统的误差估计方法显示使用RAS法将使投入产出系数的平均误差下降17.84%;而在实际应用中,使用RAS法只能使投入产出系数的平均误差下降5.92%.因此,传统的误差估计方法会非常显著低估RAS法的真实误差.从而夸大了RAS法的应用价值.     

Orthogonal Polynomials for Modified Chebyshev Measure of the First Kind

Given numbers \({n,s \in \mathbb{N}}\), \({n \geq 2}\), and the \({n}\)th-degree monic Chebyshev polynomial of the first kind \({\widehat T_n(x)}\), the polynomial system “induced” by \({\widehat T_n(x)}\) is the system of orthogonal polynomials \({\{p_{k}^{n,s} \}}\) corresponding to the modified measure \({d \sigma^{n,s}(x)=\widehat T^{2s}_n(x) d\sigma(x)}\), where \({d\sigma(x)=1/\sqrt{1-x^{2}}dx}\) is the Chebyshev measure of the first kind. Here we are concerned with the problem of determining the coefficients in the three-term recurrence relation for the polynomials \({p^{n,s}_{k}}\). The desired coefficients are obtained analytically in a closed form.     

多维广义SRLW方程的Chebyshev拟谱方法分析

考虑了一类多维的广义对称正则长波(SRLW)方程的齐次初边值问题Chebyshev拟谱逼近,构造了全离散的Chebyshev拟谱格式,给出了这种格式近似解的收敛性和最优误差估计。     

傅里叶级数的求和理论与方法

通过Dirichlet积分算子构造了一个新的积分算子Hn(f; k,x). 对于f(x)∈Cj2π, 0jk (其中k为任意自然数),Hn(f; k,x)的逼近阶达到了最佳.     

BMO函数的富里埃级数

This note announces several theorems concerning the uniform convergence of Fourier series of BMO functions and the Fourier coefficients.     

Necessary Conditions for the Convergence of Fourier Series

Various conditions have been claimed in a recent paper as necessary for the convergence of a Fourier series at a point, principally the Lebesgue summability of the series. These conditions are, however, shown to be necessary only in a particular class of functions and the error in the argument is explicated. The relevant facts about Lebesgue summability are reviewed.     

Reduced Recurrence Relations for the Chebyshev Method

In this paper, we give sufficient conditions ensuring the convergence of the Chebyshev method in Banach spaces. We use a new system of recurrence relations which simplifies those given by Kantorovich for the Newton method or those given by Candela and Marquina for the Chebyshev and Halley methods.     

Error Analysis of Clenshaw's Algorithm for Evaluating Derivatives of a Polynomial

A forward rounding error analysis is presented for the extended Clenshaw algorithm due to Skrzipek for evaluating the derivatives of a polynomial expanded in terms of orthogonal polynomials. Reformulating in matrix notation the three-term recurrence relation satisfied by orthogonal polynomials facilitates the estimate of the rounding error for the m-th derivative, which is recursively estimated in terms of the one for the (m – 1)-th derivative. The rounding errors in an important case of Chebyshev polynomial are discussed in some detail.This revised version was published online in October 2005 with corrections to the Cover Date.     

Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank--Nicolson Scheme

A parabolic problem in a separable Hilbert space is solved approximately by the projective-difference method. The problem is discretized with respect to space by the Galerkin method and with respect to time by the modified Cranck--Nicolson scheme. In this paper, we establish efficient (in time and space) strong-norm error estimates for approximate solutions. These estimates allow us to obtain the rate of convergence with respect to time of the error to zero up to the second order. In addition, the error estimates take into account the approximation properties of projective subspaces, which is illustrated for subspaces of finite element type.     

Estimates for Commutators of Orthogonal Fourier Series

In this paper we study weighted norm inequalities for the commutators[b, S_n] where b is a BMO function and S_n denotes the nth partialsum of the Fourier series relative to a system of orthogonalpolynomials on [–1, 1] with respect to general weights.Results about generalized Jacobi and Bessel Fourier series areobtained.     

聚类分析分类结果合理性考核方法

对聚类分析应用于肿瘤分型时的分类结果合理性考核方法进行探讨.该方法借助于预测强度指标,并结合中位数及95%参考值范围的概念来比较不同聚类方法得到的分类的合理性,在一定程度上弥补了聚类分析方法的应用不足.在对白血病患者分型时取得了很好的效果.