LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-06

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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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Hölder Fourier coefficients have subcritical weighted ell-2 decay

Statement

If $f \in C^{α} (T)$ with $0 < α \leq 1$ , then for every $0 < s < α$ , $\sum_{k \in Z} (1 + ∣ k ∣)^{2 s} ∣ \hat{f} (k) ∣^{2} < \infty .$

Facts & Assumptions

Given: $f \in C^{α} (T)$ and $0 < s < α$ .

[L1]

Every dyadic block has square mass $O (N^{- 2 α})$ (A dyadic Fourier-coefficient square-sum bound for Hölder functions).

Proof

technique · direct

1.1

On $2^{j} \leq ∣ k ∣ < 2^{j + 1}$ , $(1 + ∣ k ∣)^{2 s} \leq 3^{2 s} 2^{2 j s}$ ; [L1] therefore bounds that weighted block by $O (2^{- 2 j (α - s)})$ .

L1algebra

2.1

Since $α - s > 0$ , these bounds form a convergent geometric series. The $k = 0$ term is finite because $f$ is bounded and integrable.

step 1.1algebra∎

Depends on

A dyadic Fourier-coefficient square-sum bound for Hölder functions

Used by

Bernstein's absolute-convergence theorem Theorem

Dependency tree · two levels

2 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, proof of Theorem 3.3.16 (standard reference, not scraped)