Unit-direction univariate box spline ($B_m=1_{[0,1]}^{\ast m}$)

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 $\sqrt{m}$.