81 Laplace convolution

Definition. Suppose that $f$ and $g$ both satisfy $(E)$ . Then their Laplace convolution is

f\ast g(x)=\int_{0}^{x}f(x-y)g(y)\,dy.

Proposition

The Laplace convolution is:

(i) commutative, so $f\ast g=g\ast f$ ;

(ii) linear, so $(\lambda f+\mu g)\ast h=\lambda f\ast h+\mu g\ast h$ ;

(iii) multiplicative with respect to the Laplace transform, so $f\ast g$ satisfies (E) and

{\cal L}(f\ast g)(s)={\cal L}(f)(s){\cal L}(g)(s)\qquad(x>0);

(iv) associative, so $f\ast(g\ast h)=(f\ast g)\ast h$ .