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

• Paul Sablonnière
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 142)

## 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.

## Keywords

Approximation Order Minimal Support Regular Hexagon Spline Space Composition Algebra
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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