五次Hermite插值样条的最优误差界

<正> [1]对等距分划下单结点的五次插值样条作出了最优误差估计,本文将给出等距分划下五次Hermite插值样条的最优误差界. 先引入一些记号与定义. 向量(a_1,…,a_n)的弱变号数和强变号数分别记为S~-(a_1,…,a_n)和S~+(a_1,…,a_n).     

亏度为2的四次插值样条(Ⅱ)

本文是同名文章[1]的继续.所用记号均沿用[1]中规定.我们的主要目的是讨论Ⅰs,Ⅱs,Ⅲs三种类型插值法的误差阶,并给出与eeB模、连续模有关的误差估计.本文证明了:对于Ⅰs—Ⅲs型插值样条,下列不等式:     

一类自然双三次样条函数的误差估计

众所周知,二维样条插值的理论和应用研究始于1962年。1973年,C.A.Hall研究了自然双三次样条函数插值的误差估计。1981年,文献[4]将[5]的结果推广到二维情形。作者使用对双三次样条基函数的性质及偏导数的估计,将[7]的结果推广到二维情形,获得了二维样条插值中的偏导数在插值节点处的误差估计。     

一个非协调板元的误差估计

关于非协调板元的L_2—估计,文[1]曾进行了系统的研究,但对于Morely元和二个Fraeijs de Veubeke元(以下简称F.V.1元和F.V.2元)所得的结果并不是最优的。文[2]对Morely元又作了进一步的讨论,得到了最优的L_2估计及渐近最优的L_∞估计。本文将研究F.V.2元,得到了与[2]相仿的最优L_2-估计。但是,关于L_∞-估计,由于插值多     

一类二元样条的误差估计

本文通过揭示一元样条与二元样条的本质联系和构造两种局部区域上的插值函数,从而改进了[1]中S_2~1(△_(mn)~(2))上插值的误差估计结果。     

Cahn-Hilliard方程的有限元分析

建立了求解非线性发展型Cahn-Hilliard方程的有限元方法,借助于一个双调和问题的有限元投影逼近,给出了最优阶L_2模误差估计。特别对于3次Hermite型有限元,导出了L_∞模和W_∞~1模的最优阶误差估计和导数逼近的超收敛结果。     

一类基于l1模的最佳二次样条插值

本文讨论了二次样条插值的定解条件,在l_1模意义下给出了一类最佳二次样条插值的概念,以及寻找最佳二次样条插值的定解条件的方法.最后讨论了误差估计问题,并给出了实际算例.     

插值样条的敛散性与边界条件的关系

在[1]中,我们曾经讨论了五次样条插值的最优误差界和边界条件的关系,并指出了,即使结点是等距的,对某些边界条件,插值样条仍可能是发散的.本文讨论更为一般的情况,指出了对任意次多项式样条,当边界条件取某些形式时,插值样条是发散的.     

关于缺插值样条函数

在[1,2,3,4]中,我们讨论了几种缺插值样条函数.本文继续研究任意节点的缺插值样条函数,推广[1]中的结果. 在第一节中,我们讨论广义Hermite插值样条函数.通过一系列的恒等式很容易得到收敛速度的估计. 在第二节中,讨论了C~2类缺插值样条函数.建立存在性、唯一性定理,估计收敛速     

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。     

On Optimal Error Bounds for Derivatives of Interpolating Splines on a Uniform Partition

Based on Peano kernel technique, explicit error bounds (optimal for the highest order derivative) are proved for the derivatives of cardinal spline interpolation (interpolating at the knots for odd degree splines and at the midpoints between two knots for even degree splines). The results are based on a new representation of the Peano kernels and on a thorough investigation of their zero distributions. The bounds are given in terms of Euler–Frobenius polynomials and their zeros.     

五次缺插值样条的渐近性态

f(x)定义于[0,1]。将[0,1]n等分,记x_j=jh,j=0,…,n.h=1/n,且 f~(α)(x_j)=f_j~(α),j=0,…,n;α=0,1,…,5。 A.Meir和A.Sharma提出五次缺插值样条函数,即满足下面条件的函数s_n(x): (i)s_n(x)∈C~3[0,1], (ii)在区间[x_j,x_(j 1)]上(j=0,…,n-1),s_n(x)是五次多项式, (iii)s_n(x_j)=f_j,s″_n(x_j)=f″_j,j=0,…,n, (iv)s′_n(0)=f′_0,s′_n(1)=f′_n。 (1) [1]还考虑了把(1)中的(iv)换成 (iv′)s′′′_n(0)=f′′′_0,s′′′_n(1)=f′′′_n (2)的五次样条。为叙述方便,我们分别称之为(Ⅰ)型、(Ⅱ)型缺插值样条。[1]证明了(Ⅰ),(Ⅱ)型插值样条在n为奇数时是唯一存在的。[2,3,4]继续了这方面的工作,得到了一     

一类三次插值样条的余项估计

当p=α=1时,s(f,x)是通常的三次Hermite插值样条.[2,3]中的插值样条部是上述的特殊情形,本文给出了上述一般插值样条的较精确的逼近度,从中可见[2,3]中的插值样条正好都处在收敛性的临界情形.我们在讨论中利用了王兴华的基本工     

Approximation properties of the spline fit

Different topics in connection with spline fit are discussed in this paper. In particular, an example is given showing non-convergence of splines, and further some error bounds of cubic spline interpolation are proved.     

Fair upper bounds for the curvature in univariate convex interpolation

In convex interpolation the curvature of the interpolants should be as small as possible. We attack this problem by treating interpolation subject to bounds on the curvature. In view of the concexity the lower bound is equal to zero while the upper bound is assumed to be piecewise constant. The upper bounds are called fair with respect to a function class if the interpolation problem becomes solvable for all data sets in strictly convex position. We derive fair a priori bounds for classes of quadraticC ¹, cubicC ², and quarticC ³ splines on refined grids.     

Optimal Interpolation of Scattered Data on a Circular Domain with Boundary Conditions

Optimal interpolation problems of scattered data on a circular domain with two different types of boundary value conditions are studied in this paper. Closed-form optimal solutions, a new type of spline functions defined by partial differential operators, are obtained. This type of new splines is a generalization of the well-known $L_g$-splines and thin-plate splines. The standard reproducing kernel structure of the optimal solutions is demonstrated. The new idea and technique developed in this paper are finally generalized to solve the same interpolation problems involving a more general class of partial differential operators on a general region.     

The Budan-Fourier theorem for splines

The Budan-Fourier theorem for polynomials connects the number of zeros in an interval with the number of sign changes in the sequence of successive derivatives evaluated at the end-points. An extension is offered to splines with knots of arbitrary multiplicities, in which case the connection involves the number of zeros of the highest derivative. The theorem yields bounds on the number of zeros of splines and is a valuable tool in spline interpolation and approximation with boundary conditions.     

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.     

A local Lagrange interpolation method based on cubic splines on Freudenthal partitions

A trivariate Lagrange interpolation method based on cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity.

     

The derivation of cubic splines with obstacles by methods of optimization and optimal control

Summary Interpolating splines which are restricted in their movement by the presence of obstacles are investigated. For simplicity we mainly treat cubic splines which are required to be non-negative. The extension to splines of higher order and to certain other forms of obstacles is straightforward. Methods of optimization and of optimal control are used to obtain necessary optimality criteria. These criteria are applied to derive an algorithm to compute splines which are restricted to constant lower or upper bounds. There is a numerical example which illustrates the method presented.Dedicated to Günter Meinardus on the occasion of his 60th birthday