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  "body": "\n## Summary\n\nWeighted polynomial approximation measures the distance between a function and polynomials with a weight $w$, as in $\\sup_x \\abs{f(x) - p(x)} w(x)$ or in a weighted $L^p$ norm.\nOn unbounded sets such as $\\R$ the weight must decay fast enough to make polynomials have finite norm.\nA classical question, Bernstein's approximation problem, asks for which weights on $\\R$ polynomials are dense in the corresponding space of continuous functions.\nFor the weights $e^{-\\abs{x}^\\alpha}$ the answer is yes exactly when $\\alpha \\ge 1$, and [[orthogonal polynomials|8TXASARQN1]] for such weights are a main tool.\n\n## Explanations\n\n### L1 human\n\nOn a closed bounded interval, every continuous function is a uniform limit of polynomials, by the [[Weierstrass approximation theorem|0JQE130DHW]].\nOn the whole real line this fails, because a nonconstant polynomial grows without bound.\nA weight $w(x)$ that tends to $0$ quickly at infinity tames this growth, and one only asks that $(f - p) w$ be small.\nFor example, with $w(x) = e^{-x^2}$ every polynomial $p$ has $p(x) w(x) \\to 0$ as $\\abs{x} \\to \\infty$.\nHow fast the weight decays decides whether polynomials can approximate every function of the weighted space.\n",
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