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In a convenient cardinal parameter normalization, take the unit $m$-cube and project by the sum of its coordinates. Its pushforward density is the unit-direction univariate box splineEquivalently, with $B_1=1_{[0,1)}$ at its discontinuity points,Here there are $m$ convolution factors; some conventions instead index the spline by its degree $m-1$. The half-open convention for $m=1$ assigns knot values consistently and does not change the pushforward measure.

Support and positivity. A sum of $m$ numbers in $[0,1]$ lies in $[0,m]$, so the closed support is finite. For $0<x<m$, the interval of $u\in (0,1)$ with $0<x-u<m-1$ has positive length. Induction in the convolution recurrence therefore gives $B_m(x)>0$ in the interior. For $m\ge 2$, $B_m(0)=B_m(m)=0$: strict positivity includes the open support interval, not its endpoints.

Partition of unity. The translates of $B_1$ cover the line exactly once under the half-open convention. Nonnegativity permits interchange of sum and integral, and induction givesThus $\boxed{\sum _j{B}_m(x-j)=1}$, a locally finite sum. Each $B_m$ also has integral one, although partition of unity does not mean its maximum value is one.

For $m\ge 2$, repeated integration gives the explicit truncated-power formOn each unit knot interval this is a polynomial of degree at most $m-1$, and on $(0,1)$ it is $x^{m-1}/(m-1)!$, so the degree is attained. Each truncated power has $m-2$ continuous derivatives; the derivative of order $m-1$ jumps at the first knot. Therefore the dependence on the projected box dimension isFor $m=1$ the pieces are constants and the function is discontinuous at the knots. If geometric width is measured under orthogonal projection onto the unit diagonal instead, the projected coordinate is $x/\sqrt{m}$, so the physical support width is $\sqrt{m}$; its unit-mass density is $\sqrt{m}{B}_m(\sqrt{m}x)$. These are different scalings of the same cardinal spline, not different continuity or degree claims.

For a bivariate box spline, choose a spanning direction matrix $\Xi =({\xi }_1,\dots ,{\xi }_m)$ in the plane and project the unit cube by $u\mapsto \Xi u$. Define its normalization byThe support is the zonotope ${\sum }_r[0,1]{\xi }_r$, and the polynomial pieces generally have total degree $m-2$. The grid comes from the chosen direction families and their lattice translates. Two independent directions, with repetitions, give tensor-product B-splines on a parallelogram grid. Three families such as $e_1,e_2,e_1+e_2$ give a three-direction triangular grid; an affine change of basis can make it the usual equilateral triangular grid. The four families $e_1,e_2,e_1+e_2,e_1-e_2$ give the criss-cross square grid and the quadratic four-direction box spline. Changing the direction matrix and repeating directions therefore changes the grid, support and smoothness while retaining the box-projection construction.