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  "body": "\n## Summary\n\nThe convolution of two functions $f$ and $g$ on the real line is the function $(f * g)(x) = \\int_\\R f(y) g(x - y) \\, dy$.\nIt is the case $G = \\R$ of the general [[convolution|2Z29JN3VES]] on a group.\nIt models the output of a linear time-invariant system and the density of a sum of independent random variables.\nIt also describes the smoothing of a function by a kernel.\nThe [[Fourier transform|DPVM34ADZV]] turns it into a pointwise product.\n\n## Explanations\n\n### L1 human\n\nTo compute $(f * g)(x)$, flip $g$, slide it so that it sits at $x$, multiply by $f$, and take the area under the product.\nDoing this for every $x$ gives a new function, usually smoother than either of the two.\nFor example, the indicator function of $[0, 1]$ convolved with itself is a tent of height $1$ over $[0, 2]$.\nConvolving a signal with a narrow bump of area $1$ replaces each value by a local weighted average.\n",
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