DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Normalized Fourier transform on a finite abelian group

Definition

Let $G$ be a finite abelian group and $\hat{G}$ its character dual (Dual group of a finite abelian group), which is finite with $∣ \hat{G} ∣ = ∣ G ∣$ (A finite abelian group and its character dual have equal order). Put $N : = ∣ G ∣$ and $N_{R} : = ι_{R} (N)$ its canonical real image (The order $∣ G ∣$ of a finite group and the order $ord (g)$ of an element, with $ord (g) = \infty$ when no positive power of $g$ is the identity, The canonical natural $ι (n) = n \cdot 1_{F}$ of a field). Since $G$ contains its identity, $N \geq 1$ , and hence $N_{R} > 0$ (Canonical naturals are positive and strictly increasing). Its rational power $N_{R}^{- 1 / 2}$ is therefore well defined (Rational powers $a^{r}$ of a positive base, Rational powers do not depend on the representative).

For $f \in C^{G}$ , define $F_{G} f \in C^{\hat{G}}$ by $(F_{G} f) (χ) : = N_{R}^{- 1 / 2} \sum_{x \in G} f (x) \overline{χ (x)}, χ \in \hat{G} .$ Here $C^{X}$ is the space of all functions $X \to C$ (The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = {0, 1, \dots, n - 1}$ ) and the sum is the finite sum in $C$ (A finite sum in a commutative monoid indexed by an arbitrary finite set). This fixes the negative-character sign and uses the same factor $N_{R}^{- 1 / 2}$ in both directions. Unitarity is proved separately; it is not part of this definition.

Depends on

Used by

Finite cyclic discrete logarithm by Fourier sampling Algorithm
One finite abelian hidden-subgroup Fourier sample Algorithm
Finite abelian Fourier and convolution examples Example
Fourier sampling for a cyclic subgroup Example
Finite abelian Fourier inversion Theorem
Finite abelian Fourier transform is unitary Theorem
Finite abelian Parseval identity Theorem
Fourier transform of a subgroup coset indicator Theorem
Fourier transform of unnormalized convolution Theorem

Dependency tree · two levels

53 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

Ronald de Wolf, Quantum Computing: Lecture Notes, version 6 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)
John Preskill, Quantum Information, Chapter 6: Quantum Algorithms (standard reference, not scraped)