关于二元样条空间S3r^4（△＊）的维数

设△＊是任何三角剖分△的ＨＣＴ细分的三角剖分。本文建立了定义于△＊上的二元样条函数空间Ｓ３ｒ＾ｒ（△＊）的维数公式，我们的证明方法同时给出了Ｓ３ｒ＾４（△＊）的一组显示的基函数，并阐明基函数具有某种意义的局部最小支集。     

计算二维Cauchy主值积分的多元样条方法

本文采用非均匀２－型三角剖分上的样条函数空间Ｓ２＾１（Δｍｎ＾２）中的拟插值方法，构造了一类计算二维Ｃａｕｃｈｙ主值积分的数值求积公式，并对其逼近误差进行了研究。同时通过算例验证了此方法的有效性。     

Morgan-Scott剖分上样条空间S_8~4(Δ_(ms))的维数

Ｍｏｒｇｅｎ－Ｓｃｏｔ剖分上样条空间的维数依赖于剖分的几何性质，本文证明了Ｄｉｅｎｅｒ１９９０年提出的猜想对ｒ＝４是不正确的，需要修正．     

研究二元样条空间维数的Blossoming方法

本文综述了研究二元样条的Ｂｌｏｓｓｏｍｉｎｇ方法．成功地重建了平面上贯穿剖分的维数公式．而且利用这种方法，对定义在Ｍｏｒｇａｎ－Ｓｃｏｔ剖分上样条空间的维数取得了一些新的结果．     

研究多元样条的逐次分解法

本文在协调方程的基础上提出了研究多元样条的逐次分解法，并由此明了多元样条（包括多项式样条、有理样条乃至更一般的样条）在本质上是一个积分微分方和组的解。该方法具有以下优点：１）即可研究多项式样条，又可以研究有理样条乃至更一般的样条；２）即适用于三角剖分，双适用于直线剖分乃至更一般的代数曲线剖分；３）即能用于研究样条空间，又能用于研究样条环；４）可使许多问题局部化。     

二元样条的积分表示及分层三角剖分下二次样条空间的维数

本文通过引入一个积分协调条件，首次给出了二元样条的一个积分表示．文中还定义了平面单连通多边形区域的所谓分层三角剖分，并确定了此剖分下二次样条空间的维数．     

样条空间S^13（Δ^（1）mn）上的插值与逼近

本文讨论样条空间Ｓ＾１３上的插值问题，导出了一类插值条件下样条插值的存在性与唯一性结论以及计算插值样条的递推格式，其主要结论是对四阶光滑的函数，插值样条可达２阶逼近度。     

作为一元等距结点样条推广的多元样条

作者们在[4]中已经指出了给定剖分下多元B-样条存在的必要条件(1).它表明,并不是对所有的剖分都有多元B-样条存在的。人们也许以为,如同一元情况一样,只要多元B-样条存在,则它们一定组成多元样条空间的支集(即多元样条空间是所有多元B-样条所支架起来的空间)。本文以标准的三角剖分(2-单纯形)下,多元样条空间S²:=S₄²为例指出这种认识是错误的。事实上,本文定理2,3和4对这个问题已给出了明确的结论。 以上结论说明多元B-样条并不是基本的     

Morgan-Scott剖分上样条空间S^48（△ms）的维数

Morgen-Scott剖分上样条空间的维数依赖于剖分的几何性质，本文证明了Diener 1990年提出的猜想对r=4是不正确的，需要修正．     

拟矩形剖分上的样条空间的维数

矩形剖分~(记为$\Delta_{QR}$)~是指在矩形剖分~(记为$\Delta_{R}$)的基础上进行局部修改后得到的剖分,通常包括T-剖分~(记为$\Delta_{T}$)~和L-剖分~(记为$\Delta_{L}$).本文利用光滑余因子协调方法讨论了该剖分上的二元样条空间$S^\mu_k(\Delta_{QR})$的维数.在满足一定约束条件下, 得到了仅依赖于样条空间的次数,光滑度和剖分拓扑结构的显式维数公式.     

On structure of bivariate spline space of lower degree over arbitrary triangulation

The structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property. A new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces. The upper bound of the dimension of spline spaces over triangulation given by L.L. Schumaker is slightly improved via the new vertex coding method. The structure of multivariate spline spaces and over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by Luo and Wang. A kind of sufficient conditions on judging non-singularity of the and spaces over arbitrary triangulation is given, which only depends on the topological property of the triangulation. From the sufficient conditions, a triangulation strategy is presented at the end of the paper. The strategy ensures that the constructed triangulation is non-singular (or generic) for and .     

Local Lagrange Interpolation with Bivariate Splines of Arbitrary Smoothness

We describe a method which can be used to interpolate function values at a set of scattered points in a planar domain using bivariate polynomial splines of any prescribed smoothness. The method starts with an arbitrary given triangulation of the data points, and involves refining some of the triangles with Clough-Tocher splits. The construction of the interpolating splines requires some additional function values at selected points in the domain, but no derivatives are needed at any point. Given n data points and a corresponding initial triangulation, the interpolating spline can be computed in just O(n) operations. The interpolation method is local and stable, and provides optimal order approximation of smooth functions.     

A collocation method for the numerical solution of a two dimensional integral equation using a quadratic spline quasi-interpolant

In this paper, we propose an interesting method for approximating the solution of a two dimensional second kind equation with a smooth kernel using a bivariate quadratic spline quasi-interpolant (abbr. QI) defined on a uniform criss-cross triangulation of a bounded rectangle. We study the approximation errors of this method together with its Sloan’s iterated version and we illustrate the theoretical results by some numerical examples.     

QUASI-INTERPOLATING OPERATORS AND THEIR APPLICATIONS IN HYPERSINGULAR INTEGRALS

1.IntroductionSinceP.ZwartobtainedanexpressionofbivariateB--spline[2],R.--HWangandC.K.Chuihavedevelopedaseriesofresults,especially,thequasi--interpolatingoperatorsofSI(AL.)onuniformtype-2triangulationanditsapproximationproperties[']whichhavewidespreadapplicationsinMechanicsandEngineering.Furthermore,R-HWangandC.K.ChuialsoobtainedthefunctionwithminimumsupportinSI(A:)onnon-uniformtype-2triangulationandthebasisofSI(a:)[#].Inthispaperweintroducesomequasi-interpolatingoperatorsofSa(~Z~)o…     

L 1 Spline Methods for Scattered Data Interpolation and Approximation

We generalize the L ₁ spline methods proposed in [4, 5] for scattered data interpolation and fitting using bivariate spline spaces of any degree d and any smoothness r (of course, r<d) over any triangulation. Some numerical experiments are presented to illustrate the better performance of the L ₁ spline methods as compared to the minimal energy method. We include some extensions for dealing with other surface design problems.     

Morgan-Scott型剖分上的样条空间的奇异性

Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.     

Bivariate C1 Cubic Spline Spaces Over Even Stratified Triangulations

It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C ¹ cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C ¹ cubic spline spaces over a so-called even stratified triangulation.     

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function.In this paper.a coniecture on trianguation is confirmed The relation between the piecewise linear algebraiccurve and four-color conjecture is also presented.By Morgan-Scott triangulation, we will show the instabilityof Bezout number of piecewise algebraic curves. By using the combinatorial optimization method,an upper