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.

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