Smooth planar $r$-splines of degree $2r$

Details Smooth planar $r$-splines of degree $2r$ Smooth planar $r$-splines of degree $2r$

2006-08-17

arxiv.org/abs/math/0608449v1

Journal of Approximation Theory 132 (2005) 72-76

Mathematics

Numerical Analysis

6 pages, 1 figure

Scientific paper

10.1016/j.jat.2004.10.011

In \cite{as}, Alfeld and Schumaker give a formula 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 $\Delta$ (for $d \ge 3r+1$) and any triangulation (for $d\geq 3r+2$). In \cite{ss}, it was conjectured that the Alfeld-Schumaker formula actually holds for all $d \ge 2r+1$. In this note, we show that this is the best result possible; in particular, there exists a simplicial complex $\Delta$ such that for any $r$, the dimension of the spline space in degree $d=2r$ is not given by the formula of \cite{as}. The proof relies on the explicit computation of the nonvanishing of the first local cohomology module described in \cite{ss2}.

Affiliated with

Also associated with

No associations

Rating

Smooth planar $r$-splines of degree $2r$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Smooth planar $r$-splines of degree $2r$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Smooth planar $r$-splines of degree $2r$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-567744