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  "body": "\n## Definition\n\nLet $d \\le n$ and let $\\Xi = (\\xi_1, \\dots, \\xi_n)$ be a real $d \\times n$ matrix whose columns span $\\R^d$.\nThe *box spline* $M_\\Xi \\colon \\R^d \\to \\R$ is the function such that\n$$\\int_{\\R^d} f(x)\\, M_\\Xi(x)\\, dx = \\int_{[0,1]^n} f(\\Xi u)\\, du$$\nfor every continuous function $f$ on $\\R^d$ with compact support.\nEquivalently, $M_\\Xi$ is the density of the image of the uniform probability measure on $[0, 1]^n$ under the linear map $u \\mapsto \\Xi u$.\n\n## Remarks\n\nIf $n = d$, 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]].\nAdding a column gives the recursion $M_{(\\Xi, \\xi)}(x) = \\int_0^1 M_\\Xi(x - t\\xi)\\, dt$, a directional [[convolution|2Z29JN3VES]].\nExample: $d = 1$ 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.\nNon-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 $\\R^2$.\n$M_\\Xi$ is nonnegative, supported on the zonotope $\\Xi [0, 1]^n$, of integral $1$, and piecewise polynomial of degree $n - d$.\nIt has continuous derivatives of order $m - 2$, where $m$ is the least number of columns whose removal leaves a set that does not span $\\R^d$.\n",
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