Let $M = (\mathbb{R}^N, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that

(i)	$f, g : \mathbb{R}^N \to \mathbb{C}$ are each an D1921: Absolutely integrable function on $M$
(ii)	\begin{equation} X : = \{ x \in \mathbb{R}^N : y \mapsto f(x - y) g(y) \text{ is absolutely integrable on } M \} \end{equation}

The convolution of $f$ with $g$ is the D4881: Complex function \begin{equation} X \to \mathbb{C}, \quad x \mapsto \int_{\mathbb{R}^N} f(x - y) g(y) \, \ell(d y) \end{equation}