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

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

Statement

Let $0 < α \leq 1$ and let $f \in C^{α} (T)$ : $∣ f (x) - f (y) ∣ \leq C d_{T} (x, y)^{α}$ . Then there is $C_{α}$ such that, for every integer $N \geq 1$ , $\sum_{N \leq ∣ k ∣ < 2 N} ∣ \hat{f} (k) ∣^{2} \leq C_{α} C^{2} N^{- 2 α} .$

Facts & Assumptions

Given: $f, C, α, N$ as in the statement and the Fourier convention of Period-one Fourier coefficients, partial sums, and convolution on the torus.

Proof

technique · direct

1.1

Put $h = (4 N)^{- 1}$ . For $N \leq ∣ k ∣ < 2 N$ , $∣ e^{- 2 π i k h} - 1 ∣ \geq \sqrt{2}$ , since $2 π ∣ k ∣ h \in [π / 2, π)$ .

givenalgebra

1.2

For $u_{h} (x) = f (x - h) - f (x)$ , translation in the coefficient integral gives ${\hat{u}}_{h} (k) = (e^{- 2 π i k h} - 1) \hat{f} (k)$ .

givenalgebra

2.1

Finite character orthogonality applied to the block and steps 1.1--1.2 gives $2 \sum_{N \leq ∣ k ∣ < 2 N} ∣ \hat{f} (k) ∣^{2} \leq \int_{0}^{1} ∣ u_{h} (x) ∣^{2} d x .$

step 1.1step 1.2algebra

3.1

The Hölder bound gives $∣ u_{h} (x) ∣ \leq C (4 N)^{- α}$ , so step 2.1 proves the assertion.

step 2.1givenalgebra∎

Depends on

Period-one Fourier coefficients, partial sums, and convolution on the torus

Used by

Hölder Fourier coefficients have subcritical weighted ell-2 decay Lemma

Dependency tree · one level

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Sources

Loukas Grafakos, Classical Fourier Analysis, proof of Theorem 3.3.16 (standard reference, not scraped)