Box spline

Remarks

If $n = d$, then $M_\Xi = \abs{\det \Xi}^{-1}\, \mathbf{1}_{\Xi [0,1)^d}$, a multiple of the indicator function of a parallelepiped, by the change of variables formula with the determinant. Adding a column gives the recursion $M_{(\Xi, \xi)}(x) = \int_0^1 M_\Xi(x - t\xi)\, dt$, a directional convolution. Example: $d = 1$ and $\Xi = (1, 1, \dots, 1)$ with $n$ columns gives the cardinal B-spline of order $n$ with knots $0, 1, \dots, n$; for $n = 2$ this is the hat function $\min(x, 2 - x)$ on $[0, 2]$, zero elsewhere. Non-example: $d = 2$ and $\Xi$ with both columns equal to $(1, 0)^T$ fails the spanning condition; the image measure lives on a line and has no density on $\R^2$. $M_\Xi$ is nonnegative, supported on the zonotope $\Xi [0, 1]^n$, of integral $1$, and piecewise polynomial of degree $n - d$. It has continuous derivatives of order $m - 2$, where $m$ is the least number of columns whose removal leaves a set that does not span $\R^d$.