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  "body": "\n## Summary\n\nA polyharmonic spline is a sum of translates of one [[radial basis function|ZBSXSTQM2J]] $\\varphi(\\norm{x})$ on $\\R^d$ plus a low degree polynomial.\nThe radial function is $r^{2m-d}$ or $r^{2m-d} \\log r$, a multiple of the [[fundamental solution|K2NK9A8M5F]] of the iterated [[Laplacian|0E0B6RA8K0]] $\\Delta^m$ in $\\R^d$.\nPolyharmonic splines solve scattered data interpolation with minimal \"energy\", and the [[thin plate spline|P9ZPH7EC8K]] is the best known case.\n\n## Explanations\n\n### L1 human\n\nData are often measured at scattered points in the plane or in space, not on a grid.\nA polyharmonic spline puts a radially symmetric bump at each data point, adds a simple polynomial, and tunes the sizes so that the sum passes through the data.\nIn one dimension with $m = 2$ the bump is $\\abs{x}^3$, and the result is a natural cubic spline.\nIn the plane with $m = 2$ the bump is $r^2 \\log r$, which gives the thin plate spline.\n",
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