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
- Convolution of Measurable Function and Measure
- Convolution of Measures
- Young's Inequality for Convolutions
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $14.4$