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

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.

Closed form and size bounds for the Dirichlet kernel

Statement

For every integer $N \geq 0$ ,

$D_{N} (t) = 1 + 2 \sum_{k = 1}^{N} \cos (2 π k t) .$

If $t \notin Z$ , then

$D_{N} (t) = \frac{\sin ((2 N + 1) π t)}{\sin (π t)} .$

If $m \in Z$ , then $D_{N} (m) = 2 N + 1$ . In particular $D_{N}$ is even and

$∣ D_{N} (t) ∣ \leq 2 N + 1 (t \in R),$

while for $t \notin Z$ ,

$∣ D_{N} (t) ∣ \leq \frac{1}{∣ \sin (π t) ∣} .$

Facts & Assumptions

Given: An integer $N \geq 0$ and a real $t$ .

[L1]

The Dirichlet kernel is $D_{N} (t) = \sum_{∣ k ∣ \leq N} e^{2 π i k t}$ (Dirichlet and Fejer kernels).

Proof

technique · direct

1.1L1algebra

Pair the terms with indices $k$ and $- k$ in [L1]. This gives $D_{N} (t) = 1 + \sum_{k = 1}^{N} (e^{2 π i k t} + e^{- 2 π i k t}) = 1 + 2 \sum_{k = 1}^{N} \cos (2 π k t) .$ Hence $D_{N}$ is even. If $t = m \in Z$ , then every exponential equals $1$ , so $D_{N} (m) = 2 N + 1$ .

1.2L1algebra

Assume $t \notin Z$ and put $z = e^{2 π i t} \neq 1$ . Then [L1, algebra] $D_{N} (t) = z^{- N} \sum_{j = 0}^{2 N} z^{j} = z^{- N} \frac{1 - z^{2 N + 1}}{1 - z} .$ Rewriting numerator and denominator with half-angle factors yields $D_{N} (t) = \frac{e^{- π i (2 N + 1) t} - e^{π i (2 N + 1) t}}{e^{- π i t} - e^{π i t}} = \frac{\sin ((2 N + 1) π t)}{\sin (π t)} .$

2.1L1step 1.2algebra∎

The triangle inequality applied to [L1] gives $∣ D_{N} (t) ∣ \leq 2 N + 1$ for every $t$ . If $t \notin Z$ , step 1.2 and $∣ \sin ((2 N + 1) π t) ∣ \leq 1$ give $∣ D_{N} (t) ∣ \leq \frac{1}{∣ \sin (π t) ∣} .$

Depends on

Dirichlet and Fejer kernels

Used by

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