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  "body": "\n## Summary\n\nA mollifier is a smooth, nonnegative [[bump function|HS65DW0HXH]] with integral $1$, supported in the unit ball.\nRescaling it gives a family $\\varphi_\\varepsilon$ that concentrates at the origin as $\\varepsilon \\to 0$.\n[[Convolution|2Z29JN3VES]] with $\\varphi_\\varepsilon$ turns a rough function into a [[smooth function|0V33BNKZ9S]] that is close to it.\nMollifiers are the standard tool to approximate functions, prove density results and regularize distributions.\n\n## Explanations\n\n### L1 human\n\nThink of a mollifier as a small smooth blob of total mass $1$.\nConvolving a function with the blob replaces each value by a weighted average of the nearby values.\nThis blurs away corners and jumps, and the result is infinitely differentiable.\nIf the blob is made narrower, the averages are taken over smaller regions, and the blurred function gets closer to the original one.\nA standard blob on the line is $c \\, e^{-1/(1 - x^2)}$ for $\\abs{x} \u003c 1$ and $0$ elsewhere.\n",
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