Convolution

The convolution of two functions $𝑓, 𝑔 \in 𝐿 1 (ℝ 𝑛)$ is defined as #m/def/anal/fun

(𝑓 * 𝑔) (𝑡) = \int ℝ 𝑛 𝑓 (𝑡') 𝑔 (𝑡 - 𝑡') 𝑑 𝑡' = \int ℝ 𝑛 𝑓 (𝑡 - 𝑡') 𝑔 (𝑡') 𝑑 𝑡'

This forms a commutative, associative, bilinear product on integrable functions, thereby forming an $𝕂$ -monoid.

Proof

For commutativity, note

(𝑓 * 𝑔) (𝑡) = \int ℝ 𝑛 𝑓 (𝜏) 𝑔 (𝑡 - 𝜏) 𝑑 𝜏 = \int ℝ 𝑛 𝑓 (𝑡 - 𝑢) 𝑔 (𝑢) 𝑑 𝑢 = (𝑔 * 𝑡) (𝑡)

Distributivity follows from Fubini's theorem. For linearity, note

((𝜆 𝑓 + 𝜇 𝑔) * ℎ) (𝑡) = \int ℝ 𝑛 (𝜆 𝑓 (𝜏) + 𝜇 𝑔 (𝜏)) ℎ (𝑡 - 𝜏) 𝑑 𝜏 = 𝜆 \int ℝ 𝑛 𝑓 (𝜏) ℎ (𝑡 - 𝜏) 𝑑 𝜏 + 𝜇 \int ℝ 𝑛 𝑔 (𝜏) ℎ (𝑡 - 𝜏) 𝑑 𝜏 = 𝜆 (𝑓 * ℎ) (𝑡) + 𝜇 (𝑔 * ℎ) (𝑡)

and linearity in the other argument follows from commutativity.

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