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  "body": "\n## Summary\n\nConvolution combines two functions $f$ and $h$ on a group into a new function $f * h$ by averaging translates of one function, weighted by the other.\nOn $\\R^n$ it is $(f * h)(x) = \\int f(y) h(x - y) \\, dy$, and on a [[locally compact group|AHV05ZBCYW]] the integral is taken against a [[Haar measure|VY0HXD3GC4]].\nIt turns $L^1$ of the group into an algebra, it describes sums of independent random variables, and the Fourier transform turns it into pointwise multiplication.\n\n## Explanations\n\n### L1 human\n\nTo convolve a signal with a short bump, replace the value at each point by a weighted average of nearby values, with weights given by the bump.\nThe result is a smoothed version of the signal.\nFor sequences on the integers, convolving $(1, 1)$ with $(1, 2, 1)$ gives $(1, 3, 3, 1)$, exactly the coefficients of $(1 + x)(1 + 2x + x^2)$.\nSo convolution is the multiplication of polynomials and power series, written in terms of their coefficients.\n",
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