Smooth planar r-splines of degree 2r
Smooth planar r-splines of degree 2r
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2r 次平滑平面 r 样条
DOI：
10.1016/j.jat.2004.10.011
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Stefan O. Tohaneanu
中科院分区：
文献类型：
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作者：
Stefan O. Tohaneanu
Alfeld和Schumaker [Numer. Math.57（1990）651-661]给出了在平面单纯复形Δ（对于d ∈ 3r+1）和任意三角形（对于d ∈ 3r+2）的一般三角剖分上的d次和光滑度为r的分段多项式函数（样条）空间的维数的公式。在Schenck和Stiller [Mannapta Math.107（2002）43-58]中，证明了Alfred-Schumaker公式实际上对所有d 2 r +1都成立。在本文中，我们证明了这是可能的最好结果;特别地，存在一个单纯复形Δ使得对于任意r，d= 2 r的样条空间的维数不是由Alfeld和Schumaker [Numer. 57（1990）651-661]。该证明依赖于Schenck和Stillman [J. Pure Appl. Algebra 117&118（1997）535-548]中描述的第一局部上同调模的非零的显式计算。
Alfeld and Schumaker [Numer. Math. 57 (1990) 651–661] give a for mula for the dimension of the space of piecewise polynomial functions (splines) of degree d and smoothness r on a generic triangulation of a planar simplicial complex Δ (for d⩾3r+1) and any triangulation (for d⩾3r+2). In Schenck and Stiller [Manuscripta Math. 107 (2002) 43–58], it was conjectured that the Alfeld–Schumaker formula actually holds for all d⩾2r+1. In this note, we show that this is the best result possible; in particular, there exists a simplicial complex Δ such that for any r, the dimension of the spline space in degree d=2r is not given by the formula of Alfeld and Schumaker [Numer. Math. 57 (1990) 651–661]. The proof relies on the explicit computation of the nonvanishing of the first local cohomology module described in Schenck and Stillman [J. Pure Appl. Algebra 117 & 118 (1997) 535–548].