The C ^r-fundamental splines of Clough–Tocher and Powell–Sabin types for Lagrange interpolation on a three direction mesh

Abstract

Let Δ⁽¹⁾ be the uniform three direction mesh of the plane whose vertices are integer points of $\mathbb{Z}^2$.Let $\Delta _C^{(1)}$ (respectively $\Delta _P^{(1)}$ $C^r (\mathbb{R}^2 )$of degree d=3r (respectively d=3r+1 ) for r odd (respectively even) on the triangulation $\Delta _C^{(1)}$, and of degree d=2r (respectively d=2r+1) for r odd (respectively even) on the triangulation $\Delta _P^{(1)}$. Using linear combinations of translates of these splines we obtain Lagrange interpolants whose corresponding order of approximation is optimal.