---
tag: Y3WG213N0Q
kind: definition
name: Box spline
msc: [41A15, 41A63, 65D07]
defines: QMN3W21C9M
uses: [62AFX32JGT, 0RCQZ36MME, 2Z29JN3VES, 0C3CFHDN6R]
sources:
  - url: https://en.wikipedia.org/wiki/Box_spline
---

## Definition

Let $d \le n$ and let $\Xi = (\xi_1, \dots, \xi_n)$ be a real $d \times n$ matrix whose columns span $\R^d$.
The *box spline* $M_\Xi \colon \R^d \to \R$ is the function such that
$$\int_{\R^d} f(x)\, M_\Xi(x)\, dx = \int_{[0,1]^n} f(\Xi u)\, du$$
for every continuous function $f$ on $\R^d$ with compact support.
Equivalently, $M_\Xi$ is the density of the image of the uniform probability measure on $[0, 1]^n$ under the linear map $u \mapsto \Xi u$.

## Remarks

If $n = d$, then $M_\Xi = \abs{\det \Xi}^{-1}\, \mathbf{1}_{\Xi [0,1)^d}$, a multiple of the [[indicator function|0RCQZ36MME]] of a parallelepiped, by the change of variables formula with the [[determinant|0C3CFHDN6R]].
Adding a column gives the recursion $M_{(\Xi, \xi)}(x) = \int_0^1 M_\Xi(x - t\xi)\, dt$, a directional [[convolution|2Z29JN3VES]].
Example: $d = 1$ and $\Xi = (1, 1, \dots, 1)$ with $n$ columns gives the cardinal [[B-spline|62AFX32JGT]] 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 $\R^d$.