H-splines and Quasi-interpolants on a Three Directional Mesh

Abstract

Let τ (resp. τ*) be the uniform three-directional mesh of the plane generated by the vectors ${e_1} = \left( {1,0} \right),{e_2} = \left( {0,1} \right),{e_3} = \left( { - 1, - 1} \right)\;\left( {resp.\;e_1^* = (1,0),e_2^* = ( - \frac{1}{2},\frac{{\sqrt 3 }}{2}),e_3^* = ( - \frac{1}{2},\frac{{\sqrt 3 }}{2})} \right)$. Let ${\text{P}}_{n{\text{ }}}^s(\tau )$ and ${\text{P}}_{n{\text{ }}}^s(\tau *)$ be the spaces of piecewise polynomial functions of degree n and smoothness s on these meshes. There exist two interesting families of B-splines, respectively in the spaces $P_{3r + 1}^{2r}\left( \tau \right),{\text{ }}r{\text{ }} \geqslant {\text{ }}0$ and $P_{3r}^{2r - 1}\left( \tau \right),{\text{ }}r{\text{ }} \geqslant {\text{ }}1$. In the first space, B-splines with minimal support are simultaneously box-splines and H _r+1-splines, i.e., their support is the hexagon H _r+1, centered at the origin, whose sides are composed of r + 1 edges of triangles of the mesh. In the second space, there exist three types of box-splines whose supports are non regular hexagons. Generalizing examples given in [18] and [19], we construct H _r+1-splines in the space $P_{3r}^{2r - 1}\left( \tau \right)$ as linear combinations of translates of three box-splines. Then we construct various differential and discrete quasi-interpolants (QI) which have the best possible approximation order, for degrees (resp. smoothness orders) ranging from 3 to 10 (resp. from 1 to 6). Their computation is made easier thanks to the symmetry properties of H-splines. Finally, we give some examples of QI with nearly minimal infinite norms, which we call near-best quasi-interpolants.