Near-Best quasi-interpolants associated with H-splines on a three-direction mesh

Spline quasi-interpolants with best approximation orders and small norms are useful in several applications. In this paper, we construct the so-called near-best discrete and integral quasi-interpolants based on H-splines, i.e., B-splines with regular hexagonal supports on the uniform three-directional mesh of the plane. These quasi-interpolants are obtained so as to be exact on some space of polynomials, and minimize an upper bound of their infinite norms depending on a finite number of free parameters. We show that this problem has always a solution, but it is not unique in general. Concrete examples of these types of quasi-interpolants are given in the two last sections.