 Box spline

For a spanning direction matrix $\Xi =({\xi }_1,\dots ,{\xi }_m)$ in ${\mathbb{R}}^d$, the box spline is the density of the pushforward of uniform measure on $[0,1{]}^m$ under $t\mapsto \Xi t$. Equivalently,Its support is the zonotope ${\sum }_j[0,1]{\xi }_j$. It is piecewise polynomial of total degree at most $m-d$. For integer directions, splitting each integration interval in half yields its binary refinement symbolup to the monomial specifying the choice of origin. If $r$ is the smallest number of directions whose removal leaves a nonspanning set, the usual box spline smoothness criterion gives $C^{r-2}$ regularity. Repeated directions increase smoothness while enlarging support.