TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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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.

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Bernstein's absolute-convergence theorem

Statement

Assume the Axiom of Countable Choice. If $1 / 2 < α \leq 1$ and $f \in C^{α} (T)$ with respect to circular distance, then $f \in A (T)$ .

Facts & Assumptions

Given: Countable Choice, $1 / 2 < α \leq 1$ , and $f \in C^{α} (T)$ .

[L1]

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

[L2]

Weighted $ℓ^{2}$ control of exponent $s > 1 / 2$ implies $ℓ^{1}$ (Weighted ell-2 decay implies absolute convergence).

Proof

technique · direct

1.1givenchoose

Choose $s$ with $1 / 2 < s < α$ .

2.1L1L2step 1.1∎

By [L1] the weighted $ℓ^{2}$ hypothesis at this $s$ holds, and [L2] makes $(\hat{f} (k))$ summable. This is precisely $f \in A (T)$ .

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

Loukas Grafakos, Classical Fourier Analysis, Theorem 3.3.16 (standard reference, not scraped)
Michael E. Taylor, Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE, Exercise 3 (standard reference, not scraped)