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  "body": "\n## Definition\n\nLet $d \\ge 1$, let $x_1, \\dots, x_N \\in \\R^d$ be pairwise distinct, let $y_1, \\dots, y_N \\in \\R$, and let $\\varphi \\colon [0, \\infty) \\to \\R$ define a [[radial basis function|ZBSXSTQM2J]].\nLet $A$ be the $N \\times N$ matrix with entries $A_{ij} = \\varphi(\\norm{x_i - x_j})$.\nIf $A$ is invertible, the *radial basis function interpolant* of the data is\n$$s(x) = \\sum_{j=1}^{N} \\lambda_j \\, \\varphi(\\norm{x - x_j}),$$\nwhere $\\lambda = (\\lambda_1, \\dots, \\lambda_N)$ is the solution of the [[linear system|0GT4WWGWH2]] $A \\lambda = y$.\n\n## Remarks\n\nBy construction $s(x_i) = (A\\lambda)_i = y_i$, so $s$ is a [[multivariate interpolant|BM703GV34G]] of the data.\nFor the Gaussian $\\varphi(r) = e^{-(\\varepsilon r)^2}$ the matrix $A$ is a [[positive definite matrix|0KX66CZMBB]], hence invertible, for all distinct points in every dimension.\nFor conditionally positive definite functions such as the thin plate spline, a low-degree polynomial is added to $s$, with matching side conditions on $\\lambda$, to get a uniquely solvable system.\n\nExample: $d = 1$, $\\varphi(r) = r$, nodes $0, 1$, values $y_1 = 0$, $y_2 = 1$.\nThen $A = \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ and $\\lambda = (1, 0)$, so $s(x) = \\abs{x}$, and indeed $s(0) = 0$, $s(1) = 1$.\n\nNon-example: take $\\varphi(r) = r^2$ on $\\R$ and the four nodes $0, 1, 2, 3$.\nEach translate $(x - x_j)^2$ lies in the space of polynomials of degree at most $2$, which has dimension $3$.\nSo the four columns of $A$, the values of these translates at the nodes, are linearly dependent, and $A$ is not invertible.\nThen the interpolant is not defined.\n",
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