DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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 a tempered distribution with a schwartz function

Definition

For $u \in S^{'} (R^{n})$ and $φ \in S (R^{n})$ , define the scalar function

$(u * φ) (x) = ⟨ u_{y}, φ (x - y) ⟩, x \in R^{n} .$

For each fixed $x$ , the function $y \mapsto φ (x - y)$ is the reflection and translation of a Schwartz function, hence remains in $S$ by Basic operations are continuous on Schwartz space. The pairing with $u$ is therefore defined. This definition initially produces only a scalar function; its smoothness, derivative identities, polynomial growth, and regular-tempered interpretation are proved later. If $u = 0$ or $φ = 0$ , the convolution is the zero function. No convolution of two arbitrary tempered distributions is defined, and no choice axiom is used.

Convolution of a tempered distribution with a schwartz function

Definition

Depends on

Dependency tree · 2 levels

Sources