DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Finite torus fourier transform

Definition

On $V = (Z / m Z)^{2}$ , $m \geq 1$ , let $ω = e^{2 π i / m}$ and $χ_{b} (x) = ω^{b_{1} x_{1} + b_{2} x_{2}}$ . Residue representatives do not affect these values. With inner product $⟨ f, g ⟩ = m^{- 2} \sum_{x} f (x) \overline{g (x)}$ , define $\hat{f} (b) = ⟨ f, χ_{b} ⟩ = m^{- 2} \sum_{x} f (x) ω^{- b \cdot x} .$ The norm is $∥ f ∥^{2} = m^{- 2} \sum_{x} ∣ f (x) ∣^{2}$ . At $m = 1$ there is one character, the constant function one. The sign in the exponent is part of this convention.

Used by

Finite torus fourier orthogonality and affine change Lemma

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Hoory–Linial–Wigderson, Expander Graphs and Their Applications, May 2006 draft; §8.1.1 Definition8.4 and torus examples, p70. (standard reference, not scraped)