ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Fejer means of a single character

Example

Let $k \in Z$ . For every $N \geq 0$ ,

$σ_{N} e_{k} = {(1 - \frac{∣ k ∣}{N + 1})}_{+} e_{k} .$

Facts & Assumptions

Given: An integer $k$ and an integer $N \geq 0$ .

[L1]

The Cesaro means are defined by $σ_{N} f = \frac{1}{N + 1} \sum_{j = 0}^{N} S_{j} f$ (Cesaro and Abel means of a Fourier series).

Verification

technique · direct

1.1

The Fourier coefficients of $e_{k}$ vanish except at $k$ , where the coefficient is $1$ . Hence $S_{j} e_{k} = {\begin{cases} 0, & j < ∣ k ∣, \\ e_{k}, & j \geq ∣ k ∣ . \end{cases}$

L1algebra

2.1

If $∣ k ∣ > N$ , then every term in the Cesaro average from [L1] is $0$ , so $σ_{N} e_{k} = 0$ . If $∣ k ∣ \leq N$ , then exactly $N + 1 - ∣ k ∣$ of the terms equal $e_{k}$ , and therefore $σ_{N} e_{k} = \frac{N + 1 - ∣ k ∣}{N + 1} e_{k} = (1 - \frac{∣ k ∣}{N + 1}) e_{k} .$ This is the same as the displayed $(\cdot)_{+}$ formula, and for $N = 0$ it gives $σ_{0} e_{0} = e_{0}$ and $σ_{0} e_{k} = 0$ for $k \neq 0$ .

L1step 1.1algebra∎

Depends on

Cesaro and Abel means of a Fourier series

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)