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  "body": "\n## Definition\n\nLet $w \\colon \\R \\to [0, \\infty)$ be a continuous function, called the *weight*, such that $x^k w(x) \\to 0$ as $\\abs{x} \\to \\infty$ for every integer $k \\ge 0$.\nLet $C_w$ be the space of [[continuous functions|0X3WGX95FE]] $f \\colon \\R \\to \\R$ with $f(x) w(x) \\to 0$ as $\\abs{x} \\to \\infty$, with the seminorm $\\norm{f}_w = \\sup_{x \\in \\R} \\abs{f(x)} w(x)$.\nLet $\\mathcal{P}_n$ be the set of real [[polynomial functions|1G0B73WPX8]] of degree at most $n$.\n*Weighted polynomial approximation* is the approximation of $f \\in C_w$ by elements of $\\mathcal{P}_n$, measured by the weighted error\n$$E_n(f)_w = \\inf\\set{\\norm{f - p}_w : p \\in \\mathcal{P}_n} .$$\n\n## Remarks\n\nThe decay condition on $w$ ensures that every polynomial lies in $C_w$, so $E_n(f)_w$ is finite.\n$E_n(f)_w$ is the [[error of best approximation|YDN0T4MEX0]] of $f$ from $\\mathcal{P}_n$ in the seminorm $\\norm{\\cdot}_w$.\nPolynomials are dense in $C_w$ exactly when $E_n(f)_w \\to 0$ for every $f \\in C_w$; Bernstein's approximation problem asks for which weights this holds.\nThe same name is used for the weighted $L^p$ error $\\inf_p \\left( \\int_\\R \\abs{f - p}^p w \\, dx \\right)^{1/p}$, and for weights on an interval.\nExample: $w(x) = e^{-x^2}$ satisfies the decay condition, since $x^k e^{-x^2} \\to 0$ for each $k$.\nThe function $f(x) = \\abs{x}$ is continuous with $\\abs{x} e^{-x^2} \\to 0$, so $f \\in C_w$, and $E_n(f)_w \\le \\norm{f}_w = \\sup_x \\abs{x} e^{-x^2} = e^{-1/2}/\\sqrt{2}$ for every $n$, using $p = 0$.\nIn the weighted $L^2$ version with the same weight, the best constant approximation of $x^2$ is $c = \\int x^2 e^{-x^2} dx \\big/ \\int e^{-x^2} dx = 1/2$.\nNon-example: $w \\equiv 1$ fails the decay condition, and then $\\norm{p}_w = \\infty$ for every nonconstant polynomial $p$.\n",
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