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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nA *mollifier* on $\\R^n$ is a function $\\varphi \\colon \\R^n \\to \\R$ such that:\n\n1. $\\varphi$ is a [[bump function|HS65DW0HXH]], that is, [[smooth|0V33BNKZ9S]] with compact support;\n2. the support of $\\varphi$ lies in the closed unit ball $\\set{x : \\abs{x} \\le 1}$;\n3. $\\varphi(x) \\ge 0$ for all $x$;\n4. $\\int_{\\R^n} \\varphi(x) \\, dx = 1$.\n\nFor $\\varepsilon \u003e 0$ the rescaled function is $\\varphi_\\varepsilon(x) = \\varepsilon^{-n} \\varphi(x / \\varepsilon)$.\n\n## Remarks\n\nEach $\\varphi_\\varepsilon$ is again smooth and nonnegative, has integral $1$ by the change of variables $x = \\varepsilon y$, and has support in the ball of radius $\\varepsilon$.\nSome authors drop the sign condition 3 or replace condition 2 by compact support anywhere; the approximation properties remain.\n\nExample: the standard mollifier is\n$$\\varphi(x) = \\begin{cases} c \\, \\exp\\left( \\dfrac{-1}{1 - \\abs{x}^2} \\right), \u0026 \\abs{x} \u003c 1, \\\\ 0, \u0026 \\abs{x} \\ge 1, \\end{cases}$$\nwith $c \u003e 0$ chosen so that the integral is $1$.\nIt is positive inside the ball and vanishes outside, so conditions 2 and 3 hold.\nIt is smooth because $t \\mapsto e^{-1/t}$ for $t \u003e 0$, extended by $0$ for $t \\le 0$, is smooth, and $\\varphi(x) = c \\, e^{-1/t}$ with $t = 1 - \\abs{x}^2$.\n\nNon-example: the Gaussian $\\varphi(x) = \\pi^{-n/2} e^{-\\abs{x}^2}$ is smooth, positive and has integral $1$.\nIts support is all of $\\R^n$, so it fails condition 1 and condition 2.\nIt still gives an approximate identity, but convolution with it does not preserve compact support.\n",
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