Interpolation in Bivariate Spline Spaces

Abstract

Interpolation has played a crucial role in the development of univariate spline theory [11]. However in two dimensions interpolation is only known when the degree of the splines is relatively large compared to the smoothness ([9]). Let △ be a triangulation of a region, $\Omega \subseteq {\mathbb{R}^2},n,\mu \in {\mathbb{N}_o},$ and define

$$ S_n^\mu (\Delta ): = \{ s \in {c^\mu }(\Omega ):{s_{|D}} \in {\mathbb{P}_{\text{n}}}{\text{ for all triangles D of }}\Delta \} $$

, where ℙ_n denotes all polynomials of total degree n.