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

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Fourier inversion on Schwartz space

Statement

Assume countable choice. For every $f \in S (R^{n})$ and every $x \in R^{n}$ , $f (x) = \int_{R^{n}} \hat{f} (ξ) e^{2 π i x \cdot ξ} d ξ .$ The integral is absolutely convergent.

Facts & Assumptions

Given: $f \in S$ and The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

The Fourier transform preserves Schwartz space (Fourier transform acts continuously on Schwartz space).

[F2]

Schwartz functions are integrable (Schwartz derivatives are integrable).

[F3]

If $f, \hat{f} \in L^{1}$ , inversion gives its value at every Lebesgue point (L1 Fourier inversion with an integrable transform).

Proof

technique · direct

1.1

By [F1] and [F2], both $f$ and $\hat{f}$ are integrable, and the displayed integral is absolutely convergent since the exponential has modulus one. Fix $x$ . Smoothness implies continuity, so for every $ε > 0$ some $δ > 0$ gives $∣ f (x - y) - f (x) ∣ < ε$ for $∣ y ∣ < δ$ . Averaging over any ball of radius $0 < r < δ$ bounds its mean oscillation by $ε$ . Thus $x$ is a Lebesgue point with specified value $f (x)$ .

F1F2given

2.1

Apply [F3] at this arbitrary point. This proves the formula everywhere, inheriting exactly the countable-choice assumption of these three suppliers. The proof never exchanges an undamped double Fourier integral.

step 1.1F3∎

Depends on

Used by

Fourier transform is a topological automorphism of Schwartz space Corollary
Carleson tiles wave packets and tile order Definition
Carleson real line to torus transfer Lemma
Carleson single tree estimate Lemma
Wave packet model dominates the linearised carleson operator Lemma
Parseval pairing on Schwartz space Theorem

Dependency tree · two levels

22 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, MIT 18.155 (2022) (standard reference, not scraped)