Definition:Convolution of Measurable Functions

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Definition

Let $\BB^n$ be the Borel $\sigma$-algebra on $\R^n$, and let $\lambda^n$ be Lebesgue measure on $\R^n$.

Let $f, g: \R^n \to \R$ be $\BB^n$-measurable functions such that for all $x \in \R^n$:

$\ds \int_{\R^n} \map f {x - y} \map g y \rd \map {\lambda^n} y$

is finite.


The convolution of $f$ and $g$, denoted $f * g$, is the mapping defined by:

$\ds f * g: \R^n \to \R, \map {f * g} x := \int_{\R^n} \map f {x - y} \map g y \rd \map {\lambda^n} y$


Also known as

Some sources prefer the original German term Faltung (literally: folding) over convolution.


Also see


Sources