Convolution Algebra

The convolution defines a product on the linear space of integrable functions. The space of integrable functions with the product given by convolution is a commutative associative algebra without identity.

Commutativity

$$f\ast g=g\ast f$$

Associativity

$$f\ast (g\ast h)=(f\ast g)\ast h$$

Distributivity

$$f\ast (g+h)=(f\ast g)+(f\ast h)$$

Multiplicative Identity

$$f\ast \delta =f$$

where $\delta$ is the Dirac Delta Function

Inverse Element

Some function $S$ have an inverse element $S^{-}1$ for the convolution:

$$S^{-1}\ast S=\delta$$

The set of invertible distributions form an Abelian Group under convolution