C³ quartic quasi-interpolants over a 6-direction mesh

Abstract

In this work, we are interested in constructing quasi-interpolants in the space of splines S³4 (∆₆), where ∆6 designates a triangulation of a rectangular domain generated by a uniform mesh with six directions. Firstly, we will show that we can have a subspace of S³4 (∆6) containing P4 generated by the integer translates of a box spline ϕ for which we specify the B-coefficients. We also give some main properties of this box spline. Naturally, the B-coefficients of the box spline ϕ can be obtained by convolution. However, for reasons of simplicity, we propose a method based on the subdivision schemes to determine them quickly. Finally, given the importance of this triangulation, we develop some discrete and differential quasi-interpolants, and we give numerical examples.