= Box spline
{title2=$M_\Xi$}
{wiki}
For a spanning direction matrix $\Xi=(\xi_1,\ldots,\xi_m)$ 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,
$$
\int_{\mathbb R^d}M_\Xi(x)\varphi(x)\,dx=\int_{[0,1]^m}\varphi(\Xi t)\,dt.
$$
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 symbol
$$
2^d\prod_{j=1}^m\frac{1+z^{\xi_j}}2,
$$
up 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.
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