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Definition D121
Complex Lebesgue convolution

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) $$X : = \{ x \in \mathbb{R}^N : y \mapsto f(x - y) g(y) \text{ is absolutely integrable on } M \}$$
The convolution of $f$ with $g$ is the D4881: Complex function $$X \to \mathbb{C}, \quad x \mapsto \int_{\mathbb{R}^N} f(x - y) g(y) \, \ell(d y)$$
Children
 ▶ Complex Lebesgue convolution approximate identity
Results
 ▶ Complex function convolution is homogeneous to degree one ▶ Convolution is associative ▶ Convolution is commutative