基于一种新的集合插值的非凸紧集的细分概型

对A rtstein给出的度量平均的定义作了改进,给出一种新的集合插值,并基于这种新的集合插值,对相应的关于一般紧集的样条细分和插值细分分别作了研究,并给出了细分的收敛性性质.与此同时,将这种新的集合插值与基于度量平均的插值及基于M inkow sk i平均的插值分别作了比较,可以看出新的集合插值在某些方面具有更好的物理性质.     

双周期缺项插值多项式的平均逼近阶

本文首先对双周期缺项插值多项式得到了一个不等式,它是Birkhoff插值的不等式的推广.然后,应用这个不等式研究双周期缺项插值多项式平均逼近A(|z|≤1)中的函数得到了阶的估计.最后,还得到了一般的双周期缺项插值多项式收敛性的结果.     

广义Vandermonde行列式及其应用

1 广义Vandermonde行列式的定义 1966年,I.J.Schoenberg在文[1]中明确提出具有一般性的Hermite-Birkhoff插值及其插值适定性问题.而一般的Hermite-Birkhoff插值问题则未必是适定的,关于这方面目前已有许多工作,见[2]—[7].我们知道,Hermite-Birkhoff插值问题是 Hermite插值问题的推     

矩形网格上的Thiele重心型连分式混合有理插值

1引言众所周知,有理插值是非线性逼近的一种重要方法,但由于其复杂性,主要表现在有理插值问题有解是有条件的或者说有理插值问题不是总是有解的.熟知的有理插值格式(包括向量有理插值、矩阵有理插值)函数构造方法,都是假定有理插值问题有解的条件下给出的,为实际应用带来一定的困难.目前,构造有理插值常用方法之一是基于连分式给出的,应用混合方法或分块方     

Hermite插值算子在Orlicz空间内的逼近

插值算子逼近是逼近论中一个非常有趣的问题,尤其是以一些特殊的点为结点的插值算子的逼近问题很受人们的关注.研究了以第一类Chebyshev多项式零点为插值结点的Hermite插值算子在Orlicz范数下的逼近.     

具有承袭性的高阶导数有理插值算法

切触有理插值是函数逼近的一个重要内容,而降低切触有理插值的次数和解决切触有理插值函数的存在性是有理插值的一个重要问题.切触有理插值函数的算法大都是基于连分式进行的,其算法可行性是有条件的,且计算量较大.利用Newton(牛顿)多项式插值的承袭性和分段组合的方法,构造出了一种无极点且满足高阶导数插值条件的切触有理插值函数,并推广到向量值切触有理插值情形;既解决了切触有理插值函数存在性问题,又降低了切触有理插值函数的次数.最后给出误差估计,并通过数值实例说明该算法具有承袭性、计算量低、便于编程等特点.     

关于球面上的Lagrange插值

1引 言 单位球面上的插值问题一直是三元插值问题中比较受关注的部分.近年来,球面上的 Lagrange插值问题已经得到了很好地解决.例如[1]中给出了构造单位球面上的Lagrange 插值适定结点组的一种方法：添加圆周法.[2]和[3]中研究了单位球面上的多项式插值问题,给出了构造单位球面上的插值适定结点组的另外两种方法.     

关于圆弧样条逼近的若干精确估计

关于圆弧样条已有许多讨论,其中,都是针对1974年第一次CAGD国际会议上Mehlum提出的一个问题而作的,但是它们的逼近方法看起来有较大的差别,而且得到的逼近阶也很不同。本文非常简洁地给出了一般的“曲率插值”与“曲率平均插值”的圆弧样条逼近法及其比较精确的逼近度,它们不仅包含了,中的结果,更主要的是揭示了,中的内在联系:“曲率平均插值”法的逼近阶一般地高于“曲率插值”法的逼近阶。     

插值(切触)分式表的构造

用插值分式表或切触插值分式表来讨论有理插值或切触有理插值问题的一些算法的条件是比较方便的(参看[3],[6]).但关于这两个表的结构,至今未见充要的结果.为解决此问题,先引入有关术语及记号,并首先考虑有理插值的情况.     

矿藏分布型函数描述的特殊性及其插值模型

矿藏描述所提出的插值问题有二个特点,一是数据分布通常是不规则的;二是数据可能只是被讨论函数的一个泛函,所以一般的插值方法都有所不足.本文提出的插值模型能较好地处理这个问题.     

Approximation of Surfaces by Fairness Bicubic Splines

In this paper we present an approximation method of surfaces by a new type of splines, which we call fairness bicubic splines, from a given Lagrangian data set. An approximating problem of surface is obtained by minimizing a quadratic functional in a parametric space of bicubic splines. The existence and uniqueness of this problem are shown as long as a convergence result of the method is established. We analyze some numerical and graphical examples in order to prove the validity of our method.     

Construction of interpolation splines minimizing semi-norm in W_{2}^{(m,m-1)}(0,1) space

In the present paper using S.L. Sobolev’s method interpolation splines minimizing the semi-norm in a Hilbert space are constructed. Explicit formulas for coefficients of interpolation splines are obtained. The obtained interpolation spline is exact for polynomials of degree m?2 and e ^?x . Also some numerical results are presented.     

样条的机械化求解

本文在逐次分解法的基础上，给出一种样条机械化求解方法．该方法对多项式样条，有理样条乃至更一般样条的研究都是十分有效的．它适用于三角剖分，矩形剖分乃至更一般的代数曲线剖分     

样条共轭插值与L_p模误差估计

[1—5]讨论了各种类型插值样条的L_∞模最优误差估计。本文利用共轭插值样条,给出一些插值样条类的L_1模最优误差界,然后用插值空间理论导出L_p模估计的上界。 一、样条共轭插值 设n≥1并给定[0,1]上的两个分划:     

A Generalization of the Arnold-Wendland Lemma to a Modified Collocation Method for Boundary Integral Equations in ℝ3

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic periodic pseudodifferential equations in two independent variables by a modified method of nodal collocation by odd degree polynomial splines. In the one-dimensional case, our method coincides with the method of nodal collocation when odd degree polynomial splines are employed for the trial functions. The convergence analysis is based on an equivalence which we establish between our method and a nonstandard Galerkin method for an operator closely related to the given operator. This equivalence is realized through a crucial intermediate result (which we now term the Arnold-Wendland lemma) to connect the solution of central finite difference equations and that of certain nonstandard Galerkin equations. The results of this paper are genuine two-dimensional generalizations of the results obtained by ARNOLD and WENDLAND in [2] for the one-dimensional equations.     

由可逆线性系统确定的算子插值样条及其构造与连续性质

１．引言 样条理论与线性系统理论的相互联系与相互渗透,对促进样条理论与方法在随机过程最佳估计、极小能控制和数字信号处理等方面的应用有着重要的意义［１－４］．对于由一个线性微分算子确定的插值样条函数,它与线性系统及最优控制的联系是很明显的．设Ｌ（Ｄ）为线性微分算子,方程 Ｌ（Ｄ）ｙ（ｔ）＝ｕ（ｔ）总与一个能控规范型线性系统相对应, Ｌ（Ｄ）是从该系统的输出ｙ（ｔ）到输入ｕ（ｔ）的线性算子．在一个相应的再生核 Ｈｉｌｂｅｒｔ空间中,用范数｜｜Ｌ（Ｄ）ｙ（ｔ）｜｜在插值约束下的极小解定义微分算子插值样条．…     

AN INTERPOLATION METHOD BY BIVARIATE CUBIC SPLINES WITH C~2 -JOIN ON TYPE-II TRIANGULARS AT A RECTANGULAR DOMAIN

In this paper, an interpolating method for bivariate cubic splines with C2-join on type-II triangular at a rectangular domain is given, and the approximation degree, inter-polating existence and uniqueness of the cubic splines are studied.     

Uniform Approximation by the Highest Defect Continuous Polynomial Splines: Necessary and Sufficient Optimality Conditions and Their Generalisations

In this paper necessary and sufficient optimality conditions for uniform approximation of continuous functions by polynomial splines with fixed knots are derived. The obtained results are generalisations of the existing results obtained for polynomial approximation and polynomial spline approximation. The main result is two-fold. First, the generalisation of the existing results to the case when the degree of the polynomials, which compose polynomial splines, can vary from one subinterval to another. Second, the construction of necessary and sufficient optimality conditions for polynomial spline approximation with fixed values of the splines at one or both borders of the corresponding approximation interval.     

Classification of Refinable Splines

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well-known B-splines, play a key role in computer aided geometric design. So far all studies on refinable splines have focused on positive integer dilations and integer translations, and under this setting a rather complete classification was obtained in [12]. However, refinable splines do not have to have integer dilations and integer translations. The classification of refinable splines with noninteger dilations and arbitrary translations is studied in this paper. We classify completely all refinable splines with integer translations and arbitrary dilations. Our study involves techniques from number theory and complex analysis.     

Design with L-splines

We recently obtained a criterion to decide whether a given space of parametrically continuous piecewise Chebyshevian splines (i.e., splines with pieces taken from different Extended Chebyshev spaces) could be used for geometric design. One important field of application is the class of L-splines, that is, splines with pieces taken from the null space of some fixed real linear differential operator, generally investigated under the strong requirement that the null space should be an Extended Chebyshev space on the support of each possible B-spline. In the present work, we want to show the practical interest of the criterion in question for designing with L-splines. With this in view, we apply it to a specific class of linear differential operators with real constant coefficients and odd/even characteristic polynomials. We will thus establish necessary and sufficient conditions for the associated splines to be suitable for design. Because our criterion was achieved via a blossoming approach, shape preservation will be inherent in the obtained conditions. One specific advantage of the class of operators we consider is that hyperbolic and trigonometric functions can be mixed within the null space on which the splines are based. We show that this produces interesting shape effects.