CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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 L1 transform need not be integrable

Statement refuted

Every integrable function on $R$ has an integrable Fourier transform.

Facts & Assumptions

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

[F1]

The transform of $f = 1_{[- 1 / 2, 1 / 2]}$ is $\sin (π ξ) / (π ξ)$ , with value one at zero (Transform of an interval indicator).

Counterexample

1.1

Take the f in F1, whose integral and norm are one. For integer $k \geq 1$ and $ξ \in [k + 1 / 6, k + 5 / 6]$ , $∣ \sin (π ξ) ∣ \geq 1 / 2$ . Consequently $∣ \hat{f} (ξ) ∣ \geq 1 / [2 π (k + 1)]$ on an interval of length $2 / 3$ , and its integral there is at least $1 / [3 π (k + 1)]$ .

F1given

2.1

These intervals are disjoint. The sum of their lower bounds diverges: the block $2^{m} \leq k < 2^{m + 1}$ contributes at least $2^{m} / [3 π 2^{m + 1}] = 1 / (6 π)$ . Hence $\int ∣ \hat{f} ∣ = \infty$ , despite $∥ f ∥_{1} = 1$ . The failure concerns absolute integrability, not continuity or decay of the transform. Countable choice is inherited from F1 and Lebesgue interval measure.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)