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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $L^{1} (μ)$ of integrable functions

Dependency tree · 2 levels

Sources

Walter Rudin, Real and Complex Analysis, 3rd ed.