Symmetric box-splines on the An∗ lattice

Sampling and reconstruction of generic multivariate functions is more efficient on non-Cartesian root lattices, such as the BCC (Body-Centered Cubic) lattice, than on the Cartesian lattice. We introduce a new n×nn×n generator matrix A∗ that enables, in nn variables, efficient reconstruction on the non-Cartesian root lattice An∗ by a symmetric box-spline family Mr∗. A2∗ is the hexagonal lattice and A3∗ is the BCC lattice. We point out the similarities and differences of Mr∗ with respect to the popular Cartesian-shifted box-spline family MrMr, document the main properties of Mr∗ and the partition induced by its knot planes and construct, in nn variables, the optimal quasi-interpolant of M2∗.