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 nxn 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^*. A"2^* is the hexagonal lattice and A"3^* 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 M"2^*.

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