Convolution Product

A convolution product on functions G → M → 𝐺 𝑀 G\to M from a Group G 𝐺 G to a Magma M 𝑀 M is given by sums or integrals of the following form

(f1⋆f2)⁢(g)=∑h:Gf1⁢(h)⋅f2⁢(g⋅h−1)⋆subscript𝑓1subscript𝑓2𝑔subscript:ℎ𝐺⋅subscript𝑓1ℎsubscript𝑓2⋅𝑔superscriptℎ1\par(f_{1}\star f_{2})(g)=\sum_{h:G}f_{1}(h)\cdot f_{2}(g\cdot h^{-1})

Note that this works even when G 𝐺 G is a Monoid rather than a Group; we simply need to sum over Divisors of g 𝑔 g . This insight generalizes to Monoidal Functors via Day Convolution.