TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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Parseval pairing on Schwartz space

Statement

Assume countable choice. For $f, g \in S (R^{n})$ , $\int \hat{f} (ξ) \overline{\hat{g} (ξ)} d ξ = \int f (x) \overline{g (x)} d x .$ The pairing is complex-linear in the first variable. In particular $∥ \hat{f} ∥_{2} = ∥ f ∥_{2}$ .

Facts & Assumptions

Given: The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

Schwartz inversion holds everywhere with an absolutely integrable transform (Fourier inversion on Schwartz space).

[F2]

Schwartz functions are integrable and bounded (Schwartz derivatives are integrable).

[F3]

Fubini applies to absolutely integrable complex product functions on sigma-finite spaces (Fubini's theorem for L^1 functions on a sigma-finite product).

Proof

technique · direct

1.1

By [F1], $\overline{g (x)} = \int \overline{\hat{g} (ξ)} e^{- 2 π i x \cdot ξ} d ξ$ . The integrand after multiplying by $f (x)$ is jointly measurable and has absolute double integral $∥ f ∥_{1} ∥ \hat{g} ∥_{1} < \infty$ , by [F1], [F2] and product integration. Hence [F3] gives $\int f \overline{g} = \int \overline{\hat{g} (ξ)} [\int f (x) e^{- 2 π i x \cdot ξ} d x] d ξ = \int \hat{f} \overline{\hat{g}}$ . This also proves absolute integrability of the final product; the original product is integrable since $g$ is bounded and $f$ integrable.

F1F2F3given

2.1

Taking $g = f$ gives equality of the nonnegative square integrals, finite by [F2] for the input and by step 1.1 for its transform. Taking nonnegative square roots proves the norm identity. The displayed pairing is linear in its first entry and conjugate-linear in its second directly from integration and conjugation.

step 1.1F2algebra∎

Depends on

Used by

Heisenberg uncertainty and Gaussian equality Theorem
Plancherel theorem Theorem

Dependency tree · two levels

18 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

Gerald Teschl, Topics in Real and Functional Analysis (2017) (standard reference, not scraped)