in my childhood, I learned about convolution products for function over $\mathbb R$ (1). For quite a while now, I have played with polynomial rings, where also, the product is sometime called a convolution product (2). I am now discovering convolution products for arithmetic function and multiplicative functions [Apostol, Introduction to Analytic Number Theory] (3).
In case (1), the domain is $\mathbb R$, an ordered abelian group. In case (2), seeing a ring of polynomial as a group ring $\mathbb Z\langle G\rangle$, the domain is an abelain group, without ordering. In case (3), the domain seems to be the monoid $(\mathbb Z,\times)$, with an ordered by the divisibilty relation.
I don't have anything very formal, but it seems to me that there should be a general theory of convolution I don't know yet about ? Is the monoid structure the most general domain, or maybe something less structured as an acyclic graph ? Would you have lectures notes on such a theory ?
Thank you !
Just one example: A locally finite partially ordered set is a partially ordered set in which between two comparable elements there are only finitely many others. On each such set there is an "incidence algebra". Each function $f$ assigning a scalar to each interval $[a,b]=\{x : a\le x\le b\}$ is a member of the incidence algebra. The multiplication in this algebra is a sort of convolution: $$ (f*g)[a,b] = \sum_{x\,:\,a\le x\le b} f[a,x]g[x,b]. $$ The case where the partial ordering is divisibility of positive integers is well known in number theory.