Quasi-interpolants associated with H-splines on a three-direction mesh

H-splines are B-splines with regular hexagonal supports on a three-direction mesh of the plane. This family contains in particular classical box-splines which are convolution powers of the piecewise affine pyramid. In this paper, we construct discrete or integral qua si-interpolants having the best possible approximation order for the smooth functions. The construction is based on properties of hexagonal sequences and associated difference operators and of discrete Laplacian schemes.