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.

An $L^{1}$ approximate identity on $R^{n}$

Definition

An $L^{1}$ approximate identity on $R^{n}$ is a family $(K_{ε})_{ε > 0}$ of functions in $L^{1} (R^{n})$ , where

$∥ K_{ε} ∥_{1} : = \int_{R^{n}} ∣ K_{ε} (x) ∣ d x,$

such that:

$\int_{R^{n}} K_{ε} (x) d x = 1$ for every $ε > 0$ ;
there is $M < \infty$ with $∥ K_{ε} ∥_{1} \leq M$ for every $ε > 0$ ;
for every $δ > 0$ , $\int_{∣ x ∣ > δ} ∣ K_{ε} (x) ∣ d x ⟶ 0 (ε \to 0^{+}) .$

This is the general notion used later for both compactly supported mollifiers and the Gaussian family.

Depends on

The class $L^{1} (μ)$ of integrable functions

Dependency tree · 2 levels

Sources