Definition:Convolution of Measurable Functions

Definition
Let $\mathcal B^n$ 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 $\mathcal B^n$-measurable functions such that for all $x \in \R^n$:


 * $\displaystyle \int_{\R^n} f \left({x - y}\right) g \left({y}\right) \, \mathrm d \lambda^n \left({y}\right)$

is finite.

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


 * $\displaystyle f * g: \R^n \to \R, f * g \left({x}\right) := \int_{\R^n} f \left({x - y}\right) g \left({y}\right) \, \mathrm d \lambda^n \left({y}\right)$

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

Also see

 * Convolution of Measurable Function and Measure
 * Convolution of Measures
 * Young's Inequality for Convolutions