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  "body": "\n## Definition\n\nLet $d \\ge 2$, let $\\Omega \\subseteq \\R^d$, and let $X$ be a [[normed space|03J5RAKB30]] of real functions on $\\Omega$, such as $C(\\Omega)$ for compact $\\Omega$ or $L^p(\\Omega)$.\nLet $V_1, V_2, \\dots$ be subsets of $X$, called approximating families, usually finite-dimensional subspaces or sets described by $n$ parameters.\n*Multivariate approximation* is the approximation of functions $f \\in X$ of the $d$ variables by elements of the families $V_n$.\nIts quality is measured by the [[error of best approximation|YDN0T4MEX0]]\n$$E(f, V_n) = \\inf_{v \\in V_n} \\norm{f - v}_X.$$\nThe subject studies how $E(f, V_n)$ depends on $n$, on $d$ and on the smoothness of $f$.\n\n## Remarks\n\nExample: $\\Omega = [0, 1]^d$, $X = C(\\Omega)$, and $V_n$ the [[polynomials in several variables|07GNX8GCYP]] of total degree at most $n$, a space of dimension $\\binom{n + d}{d}$.\nBy the [[Stone–Weierstrass theorem|1P8D3MK79H]], $E(f, V_n) \\to 0$ for every continuous $f$.\n\nNon-example: for $d = 1$ the same question is univariate approximation, the classical case covered by the [[Weierstrass approximation theorem|0JQE130DHW]].\n\nA typical rate for a fixed dimension: if $f$ has bounded derivatives of order $s$ and $V_n$ has dimension $N$, the best linear methods give errors of order $N^{-s/d}$.\nThis rate degrades as $d$ grows.\n",
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