Symmetric box-splines on the A_n^* 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×n generator matrix A^* that enables, in n variables, efficient reconstruction on the non-Cartesian root lattice A_n^* by a symmetric box-spline family M_r^*. A₂^* is the hexagonal lattice and A₃^* is the BCC lattice. We point out the similarities and differences of M_r^* with respect to the popular Cartesian-shifted box-spline family M_r, document the main properties of M_r^* and the partition induced by its knot planes and construct, in n variables, the optimal quasi-interpolant of M₂^*.