I am relearning the convolution of two real functions $f$ and $g$ defined as follows: $$(f*g)(t)=\int_0^tf(u)g(t-u)du.$$
It satisfies some algebraic properties and also for Laplace transform, if we let $F(s)$ and $G(s)$ denote the Laplace transforms of $f(t)$ and $g(t)$, respectively, then the Laplace inverse of the product $F(s)G(s)$ is given by the function $(f*g)(t)$.
My questions:
- Does there exist an operation $f\star g$ such that if we let $F(s)$ and $G(s)$ denote the Laplace transforms of $f(t)$ and $g(t)$, respectively, then the Laplace inverse of the convolution $F(s)*G(s)$ is given by the function $(f\star g)(t)$?
- Is there a nice axiomatization of convolution using only the algebraic properties? If yes, what are some other examples of "convolutions" in other branches of mathematics? I am aware of some generalizations here but am wondering if the idea can be extended to other fields such as topology or geometry.