= Unit-direction univariate box spline
{title2=$B_m=\mathbf1_{[0,1]}^{*m}$}
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 $m\geq2$ it is $C^{m-2}$, generally not $C^{m-1}$. The recurrence $B_m(x)=\int_0^1B_{m-1}(x-u)du$ 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 $\sqrt m$.
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