Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree \(d\geq 4r+1\)
The paper considers bivariate splines. Let \(\Delta\) be a regular triangulation of a simply connected polygonal domain \(\Omega \subset {\mathbb{R}}^ 2\). Suppose the triangles of \(\Delta\) are labelled \(T^{[1]},...,T^{[M]}\) where M denotes the number of triangles. Given \(0\leq r\leq d\) we consider the linear space of bivariate splines \(S^ r_ d(\Delta)=\{s\in C^ r(\Omega):s|_{T^{[i]}}\in {\mathbb{P}}_ d\), \(i=1,...,M\}\), where \(P_ d\) is the \(\left( \begin{matrix} d+2\\ 2\end{matrix} \right)\)-dimensional linear space of bivariate polynomials of total degree d. Recently \textit{P. Alfeld} and \textit{L. L. Schumaker} [Constructing Apprxomation 3, 189-197 (1987; Zbl 0646.41008)] established the dimension of the spline spaces with \(d\geq 4r+1\), and \textit{J. Morgan} and \textit{R. Scott} in Math. Comp. 29, 736-740 (1975; Zbl 0307.65074) constructed local bases for the special case of r-1 and \(d\geq 5\). This work establishes the existence of minimally supported basis functions whenever \(d\geq 4r+1\). For the special cases \(r=1,2,3\) an explicit construction of a particular local basis is established. A fundamental tool in the analysis is the Bezier form of a bivariate polynomial.
scientific article; zbMATH DE number 4094088
scientific article