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  "body": "\n## Definition\n\nLet $d \\ge 1$ and $m \\ge 0$, and let $\\Pi_m$ be the space of [[polynomials in $d$ variables|07GNX8GCYP]] of total degree at most $m$.\nLet $x_1, \\dots, x_N \\in \\R^d$ be pairwise distinct, let $f_1, \\dots, f_N \\in \\R$, and let $\\theta \\colon [0, \\infty) \\to [0, \\infty)$ be a weight function.\nFor $x \\in \\R^d$, let $p_x \\in \\Pi_m$ minimise the [[weighted least squares|H5CGZHBR58]] error\n$$\\sum_{i=1}^{N} \\theta(\\norm{x - x_i}) \\, \\bigl(p(x_i) - f_i\\bigr)^2$$\nover $p \\in \\Pi_m$, where $\\norm{\\cdot}$ is the Euclidean norm.\nThe *moving least squares approximation* of the data is the function $\\hat f(x) = p_x(x)$, defined at every $x$ where this minimiser is unique.\n\n## Remarks\n\nThe minimiser is unique when the points $x_i$ with $\\theta(\\norm{x - x_i}) \u003e 0$ include a [[unisolvent set|5276MDSDQV]] for $\\Pi_m$.\nIn general $\\hat f$ only approximates the data; singular weights, such as in the example below, make it interpolate.\nIf the data come from a polynomial in $\\Pi_m$, then $\\hat f$ reproduces that polynomial.\n\nExample: $m = 0$ and $\\theta(r) = r^{-p}$ for $r \u003e 0$, with $p \u003e 0$, at a point $x$ that is not a data point.\nThen $p_x$ is the constant $\\sum_i \\theta_i f_i / \\sum_i \\theta_i$, where $\\theta_i = \\theta(\\norm{x - x_i})$.\nExtended by the data values at the data points, this is [[inverse distance weighting|QHN67MSH8S]].\n\nNon-example: with constant weights $\\theta = 1$ the fit is an ordinary least squares fit.\nIt gives the same polynomial $p_x = p$ at every $x$, so the result is one global polynomial.\nThe dependence of the weights on $x$, the \"moving\" feature, is lost.\n",
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