Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Bernstein's absolute-convergence theorem

Statement

If 1/2<α≤1 and f∈Cα(T) with respect to circular distance, then f∈A(T).

Facts & Assumptions

Given: 1/2<α≤1 and f∈Cα(T).

[L1]

For every 0<s<α, the Fourier coefficients of f have finite weighted ℓ2 norm of exponent s (Hölder Fourier coefficients have subcritical weighted ell-2 decay).

[L2]

Weighted ℓ2 control of exponent s>1/2 implies ℓ1 (Weighted ell-2 decay implies absolute convergence).

Proof

technique · direct
1.1

Choose s with 1/2<s<α.

givenchoose
2.1

By [L1] the weighted ℓ2 hypothesis at this s holds, and [L2] makes (f^(k)) summable. This is precisely f∈A(T).

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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