任意三角形单元上的三次Lagrange型插值逼近

本文推广了文［２］中的结果，对于任意三角形单元的三次Ｌａｇｒａｎｇｅ型插值多项式给出了原函数ｕ与被插函数Ｕ之间的误差估计     

改进Lagrange插值多项式误差上界的系数

当用Lagrange插值多项式逼近函数时,重要的是要了解误差项的性态.本文研究具有等距节点的Lagrange插值多项式,估计了Lagrange插值多项式逼近函数误差项的上界,改进了小于5次Lagrange插值多项式逼近函数误差界的系数.     

给定极点的有理函数插值序列的收敛性

设Γ∈Ｃ（１，α），α＞０．Ｇ是复平面上以Γ为边界的有界单连通区域．本文考虑了极点位于G外部，以广义Ｆａｂｅｒ-Ｄzｒｂａsｊａｎ有理函数的零点为插值结点的Ｌａｇｒａｎｇｅ插值有理函数序列对Ａ（G）和Ｅ^ｑ（Ｇ）（１＜ｑ＜＋∞）中函数的一致逼近和平均逼近阶的估计．     

核密度中含有参数的Cauchy主值积分的求积公式

在本文中，我们以二元三次复插值样条函数作为逼近工具，给出了光滑封闭曲线上的核密度中含有参数的各种Ｃａｕｃｈｙ主值积分的求积公式，误差估计及逼近性定理。     

模糊牛顿插值及模糊样条函数的表示

本文在模糊Ｌａｇｒａｎｇｅ插值的基础上，引进了模糊牛顿插值公式及其适定性定理和求法。并针对“非结点”型边界条件给出了模糊样条函数的具体表示。     

关于三次样条插值及其各阶导数的L~2误差的最佳先验界

随着样条函数的广泛应用和深入研究,三次样条插值误差的估计,在实用和理论上都具有重大的意义。本文首先给出Ⅰ型三次样条(一维与二维)的L~2误差估计,是文[1]相应部分的改进。然后证明所得结论是最佳先验界。进而把上述结果推广到Ⅲ型和Ⅳ型三次样条;最后还对Ⅲ型与Ⅳ型样条给出三阶导数误差的最佳先验界。     

线性椭圆问题最低次混合有限元方法的最优最大模误差估计

本文对线性椭圆问题的最低次混合元方法提出了构造混合元空间的充分条件,并建立了新的插值算子.据此得到了混合元解,伴随向量函数及其散度的最优最大模误差估计.     

一类五次规则HB插值样条

本文构造了C~3类五次HB插值样条空间的基函数,解决了C~3类五次规则HB插值样条的整体的构造问题,并给出了(0,2)插值误差的精确估计。     

Wilson元插值误差渐近估计

Wilson元是工程界常用的一种有限元计算方法,但在理论分析中插值误差估计的常数只知道存在,不知道具体值.本文给出了在L~2、H~1范数意义下Wilson元在参考单元和一般单元上插值误差渐近估计,导出了主要常数.这种精确的估计为有限元后验误差估计和自适应计算提供保障.     

一类新的插值结点

有界单连通区域Ｇ，其边界θＧ＝Г∈（１，α），α〉０。本计算节以广义Ｆａｂｅｒ多项式φｎ（ｚ）的零点为插值结点的Ｌａｇｒａｎｇｅ插值多项式的逼近性质，得到了它对Ａ（Ｇ↑－）中的函数的一致逼近阶和平均逼近阶的估计，并且得到了它对Ｅ＾ｐ（Ｇ）中函数的平均逼近阶的估计，还指出关于平均逼近阶的估计是不可改进的。     

距离空间中插值神经网络的误差估计

研究距离空间中的神经网络插值与逼近问题.首先引进一类广义的激活函数,用比较简洁的方法讨论距离空间中插值神经网络的存在性,然后给出插值神经网络逼近连续函数的误差估计.     

Anisotropic interpolation error estimates via orthogonal expansions

We prove anisotropic interpolation error estimates for quadrilateral and hexahedral elements with all possible shape function spaces, which cover the intermediate families, tensor product families and serendipity families. Moreover, we show that the anisotropic interpolation error estimates hold for derivatives of any order. This goal is accomplished by investigating an interpolation defined via orthogonal expansions.     

二元分形插值的拟合误差估计

A given bivariate continuous function is fitted by using a bivariate fractal interpolation function, and the error of fitting is studied in this paper. The results of error estimates are obtained in two metric cases. This provides a theoretical basis for the algorithms of fractal surface reconstruction.     

C~1连续的双3次双4次Hermite元各向异性分析

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Direct and Inverse Sobolev Error Estimates for Scattered Data Interpolation via Spherical Basis Functions

The purpose of this paper is to get error estimates for spherical basis function (SBF) interpolation and approximation for target functions in Sobolev spaces less smooth than the SBFs, and to show that the rates achieved are, in a sense, best possible. In addition, we establish a Bernstein-type theorem, where the smallest separation between data sites plays the role of a Nyquist frequency. We then use these Berstein-type estimates to derive inverse estimates for interpolation via SBFs.     

Extension Theorems for Spaces Arising from Approximation by Translates of a Basic Function

We establish extension theorems for functions in spaces which arise naturally in studying interpolation by radial basic functions. These spaces are akin in some way to the non-integer-valued Sobolev spaces, although they are considerably more general. Such extensions allow us to establish local error estimates in a way which we make precise in the introductory section of our paper. There are many other applications of these fundamental results, including improved L_p error estimates for interpolation by shifts of a single basic function, but these applications have been left to a later paper.     

Error bounds for approximation in Chebyshev points

This paper improves error bounds for Gauss, Clenshaw–Curtis and Fejér’s first quadrature by using new error estimates for polynomial interpolation in Chebyshev points. We also derive convergence rates of Chebyshev interpolation polynomials of the first and second kind for numerical evaluation of highly oscillatory integrals. Preliminary numerical results show that the improved error bounds are reasonably sharp.     

ACM元各向异性分析的Newton方法

本文将一维Lagrange插值多项式的Newton表达式推广到二维非标准的Hermite插值,给出著名板元-ACM元插值多项式的Newton表达式,由此给出ACM元对四阶和二阶椭圆问题的各向异性插值误差估计,为复杂单元的各向异性分析开辟了新的途径.