CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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Local Holder regularity implies Fourier convergence at a point

Statement

Assume the Axiom of Countable Choice.

Let $f$ be a one-period integrable function and let $x \in R$ . Suppose there are constants $C > 0$ , $α > 0$ , and $δ \in (0, 1 / 2)$ such that

$∣ f (x + t) - f (x) ∣ \leq C t^{α} and ∣ f (x - t) - f (x) ∣ \leq C t^{α}$

for every $0 < t < δ$ . Then

$S_{N} f (x) \to f (x) as N \to \infty .$

Facts & Assumptions

Given: The Axiom of Countable Choice, a one-period integrable function $f$ , a real $x$ , constants $C > 0$ and $α > 0$ , and a real $δ$ with $0 < δ < 1 / 2$ such that $∣ f (x + t) - f (x) ∣ \leq C t^{α} and ∣ f (x - t) - f (x) ∣ \leq C t^{α}$ for every $0 < t < δ$ .

[L1]

Assuming the Axiom of Countable Choice, if $\int_{0}^{δ} \frac{∣ f (x + t) + f (x - t) - 2 s ∣}{t} d t < \infty,$ then $S_{N} f (x) \to s$ (Dini pointwise convergence criterion for Fourier series).

Proof

technique · direct

1.1

For $0 < t < δ$ , the triangle inequality and the hypotheses give $∣ f (x + t) + f (x - t) - 2 f (x) ∣ \leq ∣ f (x + t) - f (x) ∣ + ∣ f (x - t) - f (x) ∣ \leq 2 C t^{α} .$

L1givenalgebra

2.1

Therefore $\int_{0}^{δ} \frac{∣ f (x + t) + f (x - t) - 2 f (x) ∣}{t} d t \leq 2 C \int_{0}^{δ} t^{α - 1} d t = \frac{2 C}{α} δ^{α} < \infty .$

step 1.1algebra

3.1

Applying [L1] with $s = f (x)$ and using step 2.1 gives $S_{N} f (x) \to f (x)$ .

L1step 2.1∎

Depends on

Dini pointwise convergence criterion for Fourier series

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

Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)
Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (standard reference, not scraped)