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.

Unnormalized convolution on a finite abelian group

Definition

Let $G$ be a finite abelian group (Group and abelian group), written additively, and let $f, g \in C^{G}$ be functions (The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = {0, 1, \dots, n - 1}$ ). Their (unnormalized) convolution is the function $(f * g) (z) : = \sum_{x \in G} f (x) g (z - x), z \in G .$ The sum is the finite sum of A finite sum in a commutative monoid indexed by an arbitrary finite set in the additive monoid of $C$ ; it is independent of any enumeration of $G$ . No factor of $∣ G ∣^{- 1}$ is included. The Fourier convolution theorem below proves the resulting normalization factor.

Depends on

Used by

Finite abelian Fourier and convolution examples Example
Fourier transform of unnormalized convolution Theorem

Dependency tree · two levels

17 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)