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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $\\norm{\\cdot}$ be the Euclidean norm on [[Euclidean space|0XPR1E85W0]] $\\R^d$.\nA *radial basis function* is a [[function|0EPAT51DXQ]] $\\Phi \\colon \\R^d \\to \\R$ of the form\n$$\\Phi(x) = \\varphi(\\norm{x}),$$\nfor some function $\\varphi \\colon [0, \\infty) \\to \\R$.\nFor a centre $c \\in \\R^d$, the translate $x \\mapsto \\varphi(\\norm{x - c})$ depends only on the [[Euclidean distance|0NPPGHQNXQ]] from $x$ to $c$.\n\n## Remarks\n\nCommon choices of $\\varphi$, with a shape parameter $\\varepsilon \u003e 0$:\n\n1. Gaussian: $\\varphi(r) = e^{-(\\varepsilon r)^2}$;\n2. multiquadric: $\\varphi(r) = \\sqrt{1 + (\\varepsilon r)^2}$;\n3. inverse multiquadric: $\\varphi(r) = 1 / \\sqrt{1 + (\\varepsilon r)^2}$;\n4. polyharmonic splines: $\\varphi(r) = r^k$ for odd $k$ and $\\varphi(r) = r^k \\log r$ for even $k$, with $\\varphi(0) = 0$.\n\nExample: $\\Phi(x) = e^{-\\norm{x}^2}$ on $\\R^2$ is radial: $\\Phi(x_1, x_2) = e^{-(x_1^2 + x_2^2)}$ only depends on $\\norm{x}$.\nFor instance $\\Phi(1, 0) = \\Phi(0, 1) = e^{-1}$.\n\nNon-example: $\\Psi(x_1, x_2) = e^{-(x_1^2 + 2 x_2^2)}$ is not radial: $\\norm{(1, 0)} = \\norm{(0, 1)} = 1$, but $\\Psi(1, 0) = e^{-1} \\ne e^{-2} = \\Psi(0, 1)$.\n",
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