Box spline ($M_{\Xi }$)

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.

The unit-direction univariate box spline is the density of the sum of $m$ independent uniform unit-interval coordinates. It is the order-$m$ Cardinal B-spline: its closed support is $[0,m]$, it is strictly positive on the interior, and its polynomial degree is $m-1$. For $m\ge 2$ it is $C^{m-2}$, generally not $C^{m-1}$. The recurrence $B_m(x)={\int }_0^1{B}_{m-1}(x-u)du$ proves its support and positivity. Starting with a half-open unit interval, integrating the translate partition recursively proves ${\sum }_{j\in \mathbb{Z}}{B}_m(x-j)=1$. Orthogonal projection onto the unit diagonal rescales support width from $m$ to $\sqrt{m}$.

This box spline is the density of the image of the uniform unit four-cube under the indicated direction matrix. Its polynomial pieces have total degree $4-2=2$. Removing three directions is necessary to leave a nonspanning set, giving $C^1$ continuity. Its knot arrangement is a criss-cross triangulation of a square lattice. Its refinement mask is $\frac{1}{4}(1+z)(1+w)(1+zw)(1+z/w)$, up to the monomial that fixes the origin.

Take two copies of each direction $(1,0),(0,1),(1,1),(1,-1)$. The resulting box spline has total polynomial degree six and $C^4$ continuity: there are eight directions and a rank-deficient remaining set has at most two, so the removal number is six. With a centred origin its support is the zonotopeIts centred binary symbol isThis symbol factors as $A(z,w)A(zw,z/w)$ with $A=(2+z+z^{-1})(2+w+w^{-1})/8$, giving an efficient quincunx subdivision implementation. The box spline criterion is also described in the primary research paper 4–8 Subdivision by Velho and Zorin.