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

How statement and proof provenance work

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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.

Depends on

Used by

Tempered convolution is smooth with polynomial growth Theorem

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Semyon Dyatlov, Lecture notes for 18.155 (2022) (standard reference, not scraped)
Radu Gelca, Functional Analysis (standard reference, not scraped)