Convolution on the line

Summary

The convolution of two functions $f$ and $g$ on the real line is the function $(f * g)(x) = \int_\R f(y) g(x - y) \, dy$. It is the case $G = \R$ of the general convolution on a group. It models the output of a linear time-invariant system and the density of a sum of independent random variables. It also describes the smoothing of a function by a kernel. The Fourier transform turns it into a pointwise product.