DefinitionDefinition: AI-adaptedProof: Not applicableverified 2026-09-23 (gpt-6-sol)

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.

Dirichlet and Fejer kernels

Definition

For $N \geq 0$ , the Dirichlet kernel on $T$ is

$D_{N} (t) : = \sum_{∣ k ∣ \leq N} e_{k} (t) .$

The Fejer kernel is the arithmetic mean

$F_{N} (t) : = \frac{1}{N + 1} \sum_{j = 0}^{N} D_{j} (t) .$

Equivalently,

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

so $D_{N}$ is real-valued and even. Also

$\int_{0}^{1} D_{N} (t) d t = 1,$

where this is the ordinary Riemann integral of the continuous finite sum. Every nonconstant character has Riemann integral $0$ over one period. Under Countable Choice it agrees with the torus Lebesgue integral in Period-one Fourier coefficients, partial sums, and convolution on the torus.

Depends on

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

Used by

Dependency tree · two levels

8 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)