Convolution

Summary

Convolution 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. On $\R^n$ it is $(f * h)(x) = \int f(y) h(x - y) \, dy$, and on a locally compact group the integral is taken against a Haar measure. It 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.