ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Simple map
Simple function
Measurable simple complex function
Simple integral
Unsigned basic integral
P-integrable basic function
Set of P-integrable complex Borel functions
Convolution
Definition D121
Complex Lebesgue convolution
Formulation 3
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}
Children
Complex Lebesgue convolution approximate identity
Results
Complex function convolution is homogeneous to degree one
Convolution is associative
Convolution is commutative