1. The product of two power series/polynomials is
The coefficients given by
is sometimes called the Cauchy product. This is a convolution.
2. Let be a finite group and be the group algebra with complex coefficients. Let
be two elements in . The product is
When and are treated as functions ,
is a convolution. In fact, is an example of convolution algebra .
3. Let and . is a groupoid: not every pair of elements in can be composed, can only be composed with when . In such case, Let us consider the groupoid convolution algebra as above. The convolution in this case is then
If we rewrite this in another format, it should look more familiar:
This is matrix multiplication.