structural:scholar_edition
Algebraic properties

The convolution defines a product on the linear space of integrable functions. This product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative associative algebra without identity (Strichartz 1994, §3.3). Other linear spaces of functions, such as the space of continuous functions of compact support, are closed under the convolution, and so also form commutative associative algebras.

Proof (using convolution theorem):

q (t) ⟺ F Q (f) = R (f) S (f) {\displaystyle q(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(f)=R(f)S(f)}

q (− t) ⟺ F Q (− f) = R (− f) S (− f) {\displaystyle q(-t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(-f)=R(-f)S(-f)}

q (− t) = F − 1 { R (− f) S (− f) } = F − 1 { R (− f) } ∗ F − 1 { S (− f) } = r (− t) ∗ s (− t) {\displaystyle {\begin{aligned}q(-t)&={\mathcal {F}}^{-1}{\bigg \{}R(-f)S(-f){\bigg \}}\\&={\mathcal {F}}^{-1}{\bigg \{}R(-f){\bigg \}}*{\mathcal {F}}^{-1}{\bigg \{}S(-f){\bigg \}}\\&=r(-t)*s(-t)\end{aligned}}}