ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.

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The Dirichlet kernel at zero and away from zero

Example

For each $N \geq 1$ ,

$D_{N} (0) = 2 N + 1,$

while

$D_{N} ⁣ (\frac{1}{4 N + 2}) = \frac{1}{\sin (π / (4 N + 2))} > 0, D_{N} ⁣ (\frac{3}{4 N + 2}) = - \frac{1}{\sin (3 π / (4 N + 2))} < 0.$

So the Dirichlet kernel has a removable peak at $0$ and already changes sign at two explicit nearby points.

Facts & Assumptions

Given: An integer $N \geq 1$ .

[L1]

The Dirichlet kernel is $D_{N} (t) = \sum_{∣ k ∣ \leq N} e_{k} (t)$ (Dirichlet and Fejer kernels).

[L2]

For $t \notin Z$ , $D_{N} (t) = \frac{\sin ((2 N + 1) π t)}{\sin (π t)},$ and $D_{N} (0) = 2 N + 1$ (Closed form and size bounds for the Dirichlet kernel).

Verification

technique · direct

1.1L1L2algebra

At $t = 0$ , [L1] gives $D_{N} (0) = \sum_{∣ k ∣ \leq N} e_{k} (0) = \sum_{∣ k ∣ \leq N} 1 = 2 N + 1,$ which agrees with the removable value recorded in [L2].

2.1L2algebra∎

The numbers $(4 N + 2)^{- 1}$ and $3 (4 N + 2)^{- 1}$ are not integers, so [L2] applies. Since $(2 N + 1) π / (4 N + 2) = π / 2$ and $3 (2 N + 1) π / (4 N + 2) = 3 π / 2$ , the numerator is respectively $1$ and $- 1$ . The denominators are positive because their angles lie in $(0, π)$ . Hence the two displayed signs follow.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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