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  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] with a left [[Haar measure|VY0HXD3GC4]] $\\mu$.\nLet $f, h \\colon G \\to \\C$ be functions in [[$L^1(G)$|SASP14MA1Z]], the space of $\\mu$-integrable functions.\nThe *convolution* of $f$ and $h$ is the function $f * h$ on $G$ given by\n$$(f * h)(x) = \\int_G f(y) \\, h(y^{-1} x) \\, d\\mu(y)$$\nfor every $x \\in G$ at which the integral converges absolutely.\n\n## Remarks\n\nThe integral converges absolutely for almost every $x$, and $f * h$ lies in $L^1(G)$ with $\\norm{f * h}_1 \\le \\norm{f}_1 \\norm{h}_1$.\nConvolution is associative, and it is commutative when $G$ is abelian.\n\nExample: $G = (\\R, +)$ with Lebesgue measure, $f = h = \\mathbf{1}_{[0, 1]}$.\nThen $(f * h)(x) = \\int_\\R \\mathbf{1}_{[0,1]}(y) \\mathbf{1}_{[0,1]}(x - y) \\, dy$ is the length of $[0, 1] \\cap [x - 1, x]$.\nThis equals $x$ for $0 \\le x \\le 1$, $2 - x$ for $1 \\le x \\le 2$, and $0$ otherwise: a triangle function.\n\nNon-example: for $f = h = 1$ on $\\R$, the defining integral $\\int_\\R 1 \\, dy$ diverges at every $x$, so the convolution is not defined; the hypothesis $f, h \\in L^1$ fails.\n\nOn $(\\Z, +)$ with counting measure, $(f * h)(n) = \\sum_{m \\in \\Z} f(m) h(n - m)$, the Cauchy product of coefficient sequences.\nThe convolution of probability distributions is the special case of measures, which describes sums of independent random variables.\n",
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