Has the set of functions $\mathbb{F}_q \to \mathbb{Z}$ endowed with pointwise addition and additive/multiplicative convolution been studied?
Does anyone know a reference or keywords to search for?
Edit:
This construction is an analogon to integral group rings, which are (for finite groups $G$) just the functions $G \to \mathbb{Z}$. In fact they are related. With pointwise addition and additive convolution $\mathbb{F}_p \to \mathbb{Z}$ with $p$ prime is the same as the group ring $C_p \to \mathbb{Z}$. I'm interested in what\whether the multiplicative convolution adds to the construction of integral group rings and what happens in the prime power case.
A function $k: \mathbb{F}_q \to \mathbb{Z}$ can be thought of as a formal sum $k = \sum_{f \in \mathbb{F}_q} a_f f$ with $a_f \in \mathbb{Z}$.
With $l = \sum_{f \in \mathbb{F}_q} b_f f$ we would have:
- Additive convolution: $k \star_a l = \sum_{f \in \mathbb{F}_q}(\sum_{g+h=f} \;a_g b_h)f$.
- Multiplicative convolution: $k \star_m l = \sum_{f \in \mathbb{F}_q}(\sum_{gh=f} \;a_g b_h)f$.