Name
Abstract | ||
|---|---|---|
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]. |
Year | DOI | Venue |
|---|---|---|
2005 | 10.1016/j.jat.2004.10.011 | Journal of Approximation Theory |
Keywords | DocType | Volume |
primary 13D40,secondary 52B20 | Journal | 132 |
Issue | ISSN | Citations |
1 | Journal of Approximation Theory 132 (2005) 72-76 | 3 |
PageRank | References | Authors |
0.57 | 1 | 1 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Stefan O. Tohaneanu | 1 | 15 | 5.03 |