Convolution on the half-line

Summary

The convolution on the half-line combines two functions $f, g$ on $[0, \infty)$ into $(f * g)(t) = \int_0^t f(\tau) g(t - \tau) \, d\tau$. It is the convolution on the line of the two functions extended by zero to negative times. The Laplace transform turns it into a pointwise product. It is also the product of the Mikusiński operational calculus.