Spaces of bivariate cubic and quartic splines on type-1 triangulations
Spaces of bivariate cubic and quartic splines on type-1 triangulations
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DOI：
10.1016/0022-247x(84)90118-5
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发表时间：
1984-07
影响因子：
1.3
通讯作者：
C. Chui;Ren-hong Wang
中科院分区：
文献类型：
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作者：
C. Chui;Ren-hong Wang
出于理论和计算上的原因，通常希望将样条函数表示为B-样条线的线性组合(参见。De Boor[11]。在网格划分为“均匀”的二元设置中，可以考虑Fredrickson[I‘]的三角样条线性组合，Sablonniere[8]构造的等边三角形上支撑的样条线，或者更一般地，由De Boor和Hollig[2]引入的盒样条。三个重要的代数问题立即出现：(1)这些局部支承的样条函数的平移是否足以生成相同次数且满足相同光滑性条件的所有样条函数？(2)如果是这样的话，我们可以用什么方便的方法从这组可能线性相关的平移函数中选择一个基？(3)如果不是，什么函数应该包括在生成集中，最好是尽可能光滑且具有“小”支撑点的函数？本文的目的是回答L三角剖分上的三次和四次样条线的这些问题。我们从必要的符号开始。由两个变量的总次数为d的所有多项式组成的空间将由P表示。设m和n为正整数。分区
For both theoretical and computational reasons, it is usually desirable to represent spline functions as linear combinations of B-splines (cf. de Boor [11). In the bivariate setting when the grid partition is “uniform” one could consider linear combinations of triangular splines of Fredrickson [i’], splines supported on equilateral triangles constructed by Sablonniere [8], or, more generally, the box splines introduced by de Boor and Hollig [2]. Three important algebraic questions arise immediately:(1) Are the translates of these locally supported spline functions enough to generate all spline functions of the same degree and satisfying the same smoothness conditions?(2) If so, in what convenient ways can we choose a basis from this possibly linearly dependent set of translates?(3) If not, what functions, preferably as smooth as possible and having “small” supports, should be included in the generating set? The purpose of this paper is to answer these questions for cubic and quartic splines on type-l triangulations. We begin with the necessary notation. The space of all polynomials of total degree d in two variables will be denoted by P,. Let where m and n are positive integers. The partition