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  "body": "\n## Definition\n\nLet $d \\ge 1$ and $m \\ge 1$ be integers with $2m \u003e d$.\nDefine $\\varphi \\colon [0, \\infty) \\to \\R$ by\n$$\\varphi(r) = \\begin{cases} r^{2m-d} \\log r \u0026 \\text{if } d \\text{ is even}, \\\\ r^{2m-d} \u0026 \\text{if } d \\text{ is odd}, \\end{cases}$$\nwith $\\varphi(0) = 0$.\nLet $\\Pi_{m-1}$ be the space of polynomials on $\\R^d$ of total degree at most $m - 1$.\nA *polyharmonic spline* of order $m$ with distinct centres $x_1, \\dots, x_N \\in \\R^d$ is a function\n$$s(x) = \\sum_{i=1}^{N} w_i\\, \\varphi(\\norm{x - x_i}) + p(x),$$\nwhere $p \\in \\Pi_{m-1}$ and the real weights satisfy $\\sum_{i=1}^{N} w_i\\, q(x_i) = 0$ for every $q \\in \\Pi_{m-1}$.\n\n## Remarks\n\nSuppose the centres are unisolvent for $\\Pi_{m-1}$, that is, the only $q \\in \\Pi_{m-1}$ vanishing at all $x_i$ is $0$.\nThen for any data $y_1, \\dots, y_N$ there is exactly one polyharmonic spline with $s(x_i) = y_i$.\nThis is [[radial basis function interpolation|XZ33P3HBB9]] with the [[radial basis function|ZBSXSTQM2J]] $\\varphi$.\nUp to a constant factor, $\\varphi(\\norm{x})$ is a [[fundamental solution|K2NK9A8M5F]] of $\\Delta^m$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]], so $s$ satisfies $\\Delta^m s = 0$ away from the centres.\nThe interpolant minimizes the seminorm $\\sum_{\\abs{\\alpha} = m} \\frac{m!}{\\alpha!} \\int_{\\R^d} (\\partial^\\alpha f)^2$ among suitable functions with $f(x_i) = y_i$.\nExample: $d = 1$, $m = 2$ gives $\\varphi(r) = r^3$ and $s(x) = \\sum w_i \\abs{x - x_i}^3 + a + bx$ with $\\sum w_i = \\sum w_i x_i = 0$; this is a natural cubic spline.\nNon-example: in the same setting, $s(x) = \\abs{x}^3$ (one centre $0$, $w_1 = 1$) violates the side condition $\\sum w_i = 0$, and indeed it is not linear outside the centres.\nFor $d = 2$, $m = 2$ one gets $\\varphi(r) = r^2 \\log r$, the thin plate spline.\n",
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