DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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Convolution of two functions on $R^{n}$

Definition

Let $f, g : R^{n} \to C$ be measurable functions. When, for a given $x \in R^{n}$ , the function

y ⟼ f (x - y) g (y)

is measurable and integrable on $R^{n}$ , the convolution of $f$ and $g$ at $x$ is

(f * g) (x) : = \int_{R^{n}} f (x - y) g (y) d y .

For $L^{1}$ classes, later items make the representative convention explicit.

Depends on

The class $L^1(\mu)$ of integrable functions

Used by

Dependency tree · two levels

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Sources

Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)

Convolution of two functions on RnRn