Plancherel theorem for locally compact abelian groups

E876155

theorem in harmonic analysis

The Plancherel theorem for locally compact abelian groups is a fundamental result in harmonic analysis that identifies the Fourier transform as a unitary isomorphism between an L²-space on the group and an L²-space on its dual group, preserving inner products and norms.

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All labels observed (7)

Label	Occurrences
Plancherel theorem	8
Plancherel's theorem	2
Fourier transform is an isometry on L2(G)	1
Plancherel formula	1
Plancherel theorem for locally compact abelian groups canonical	1
Plancherel theorem for the circle group	1
Plancherel theorem for the real line	1

How this entity was disambiguated

This entity first appeared as the object of triple T10641466 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Plancherel theorem for locally compact abelian groups
Context triple: [Plancherel theorem for real reductive groups, generalizes, Plancherel theorem for locally compact abelian groups]

A. Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
B. Introduction to Abstract Harmonic Analysis
Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
C. Pontryagin duality
Pontryagin duality is a fundamental theorem in harmonic analysis and topological group theory that establishes a duality between locally compact abelian groups and their groups of continuous characters.
D. Bochner theorem on characteristic functions
The Bochner theorem on characteristic functions is a fundamental result in probability theory and harmonic analysis that characterizes which functions are Fourier transforms of probability measures by requiring them to be positive-definite, continuous, and normalized at zero.
E. Harmonic Analysis and the Theory of Probability
Harmonic Analysis and the Theory of Probability is a seminal mathematical monograph that connects Fourier-analytic methods with probabilistic concepts, helping to lay the foundations of modern probability theory.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Plancherel theorem for locally compact abelian groups
Target entity description: The Plancherel theorem for locally compact abelian groups is a fundamental result in harmonic analysis that identifies the Fourier transform as a unitary isomorphism between an L²-space on the group and an L²-space on its dual group, preserving inner products and norms.

A. Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
B. Introduction to Abstract Harmonic Analysis
Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
C. Pontryagin duality
Pontryagin duality is a fundamental theorem in harmonic analysis and topological group theory that establishes a duality between locally compact abelian groups and their groups of continuous characters.
D. Bochner theorem on characteristic functions
The Bochner theorem on characteristic functions is a fundamental result in probability theory and harmonic analysis that characterizes which functions are Fourier transforms of probability measures by requiring them to be positive-definite, continuous, and normalized at zero.
E. Harmonic Analysis and the Theory of Probability
Harmonic Analysis and the Theory of Probability is a seminal mathematical monograph that connects Fourier-analytic methods with probabilistic concepts, helping to lay the foundations of modern probability theory.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	theorem in harmonic analysis ⓘ
appliesTo	L2-spaces ⓘ locally compact abelian groups ⓘ
assumes	Fourier transform initially defined on L1(G) ∩ L2(G) ⓘ linked to: Fourier analysis on groups G is a locally compact abelian group ⓘ
category	result about unitary representations ⓘ
codomainSpace	L2(Ĝ) ⓘ
conclusion	Fourier transform extends uniquely to all of L2(G) ⓘ linked to: Fourier transform is defined almost everywhere for L2-functions
domainSpace	L2(G) ⓘ
ensures	Parseval identity for L2-functions on G ⓘ
field	abstract harmonic analysis ⓘ functional analysis ⓘ harmonic analysis ⓘ
generalizes	Plancherel theorem for finite abelian groups ⓘ Plancherel theorem for the circle group ⓘ linked to: Plancherel theorem for locally compact abelian groups Plancherel theorem for the real line ⓘ linked to: Plancherel theorem for locally compact abelian groups
holdsFor	second countable locally compact abelian groups ⓘ
implies	Fourier transform is defined almost everywhere for L2-functions ⓘ equality of L2-norms of a function and its Fourier transform ⓘ
importance	fundamental result in abstract harmonic analysis ⓘ
involvesConcept	Fourier transform ⓘ Haar measure ⓘ L2-norm ⓘ Pontryagin duality ⓘ dual group ⓘ inner product ⓘ isometry ⓘ unitary isomorphism ⓘ unitary operator ⓘ
mathematicalArea	analysis on topological groups ⓘ
property	Fourier transform is an isometry on L2(G) ⓘ linked to: Plancherel theorem for locally compact abelian groups Fourier transform is surjective from L2(G) onto L2(Ĝ) ⓘ preserves L2-norms ⓘ preserves inner products ⓘ
relatedTo	Fourier inversion theorem ⓘ Parseval identity ⓘ linked to: Parseval's theorem Pontryagin duality theorem ⓘ linked to: Pontryagin duality
relates	L2(G) with L2(Ĝ) ⓘ
requires	choice of Haar measure on G ⓘ corresponding Haar measure on the dual group Ĝ ⓘ
statement	the Fourier transform extends to a unitary operator from L2(G) onto L2(Ĝ) ⓘ
typicalNotation	ℱ: L2(G) → L2(Ĝ) ⓘ
usedIn	Fourier analysis on groups ⓘ representation theory of abelian groups ⓘ signal processing on LCA groups ⓘ

How these facts were elicited

Referenced by (15)

Full triples — surface form annotated when it differs from this entity's canonical label.

Plancherel theorem for real reductive groups → generalizes → Plancherel theorem for locally compact abelian groups ⓘ

Fourier inversion theorem → isRelatedTo → Plancherel theorem ⓘ