Let $k$ be a commutative unital ring, $(C,\Delta)$ a (counital) $k$-coalgebra and $(A,m)$ an associative (unital) $k$-algebra. Then the set of linear maps

$\mathrm{Hom}_k(C,A)$

has a structure of an associative (unital) algebra, called convolution algebra, in which the product of two linear maps $f,g$ is given by

$f\star g = m\circ(f\otimes g)\circ\Delta.$

Of a group

Given a finite group$G$ and a ring$R$, the space of functions$C(G,R)$ inherits the convolution product defined by