Paley–Wiener theorem

E898512

mathematical theorem theorem in harmonic analysis

The Paley–Wiener theorem is a fundamental result in harmonic analysis that characterizes which functions arise as Fourier transforms of compactly supported functions (or distributions), linking analytic properties of entire functions with support properties in the original domain.

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All labels observed (7)

Label	Occurrences
Paley–Wiener–Schwartz theorem	2
Paley–Wiener theorem canonical	1
Paley–Wiener theorem for distributions	1
Paley–Wiener theorem for the Fourier transform on R^n	1
Paley–Wiener theorem for the Fourier transform on locally compact abelian groups	1
Paley–Wiener type theorems	1
classical Paley–Wiener theorem on ℝⁿ	1

How this entity was disambiguated

This entity first appeared as the object of triple T10992118 — 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: Paley–Wiener theorem
Context triple: [Fourier inversion theorem, isRelatedTo, Paley–Wiener theorem]

A. Riesz–Thorin interpolation theorem
The Riesz–Thorin interpolation theorem is a fundamental result in functional analysis that provides bounds for linear operators between Lᵖ spaces by interpolating their behavior between two known endpoint estimates.
B. Wiener–Khinchin theorem
The Wiener–Khinchin theorem is a fundamental result in signal processing and probability theory that relates a wide-sense stationary random process’s autocorrelation function to its power spectral density via the Fourier transform.
C. Littlewood–Paley theory
Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
D. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
E. Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
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: Paley–Wiener theorem
Target entity description: The Paley–Wiener theorem is a fundamental result in harmonic analysis that characterizes which functions arise as Fourier transforms of compactly supported functions (or distributions), linking analytic properties of entire functions with support properties in the original domain.

A. Riesz–Thorin interpolation theorem
The Riesz–Thorin interpolation theorem is a fundamental result in functional analysis that provides bounds for linear operators between Lᵖ spaces by interpolating their behavior between two known endpoint estimates.
B. Wiener–Khinchin theorem
The Wiener–Khinchin theorem is a fundamental result in signal processing and probability theory that relates a wide-sense stationary random process’s autocorrelation function to its power spectral density via the Fourier transform.
C. Littlewood–Paley theory
Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
D. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
E. Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in harmonic analysis ⓘ
appliesTo	L1 functions ⓘ L2 functions ⓘ Schwartz space ⓘ tempered distributions ⓘ
characterizes	entire functions of exponential type ⓘ entire functions that are Fourier transforms of compactly supported functions ⓘ
concerns	Fourier transform ⓘ entire functions on C^n ⓘ entire functions on the complex plane ⓘ support of a function ⓘ
context	Fourier transform on R ⓘ Fourier transform on R^n ⓘ
describes	Fourier transforms of compactly supported distributions ⓘ Fourier transforms of compactly supported functions ⓘ
field	Fourier analysis ⓘ complex analysis ⓘ harmonic analysis ⓘ
givesConditionOn	exponential type of entire functions ⓘ growth of entire functions along imaginary axis ⓘ support of original function in real domain ⓘ
hasVariant	Paley–Wiener theorem for distributions ⓘ linked to: Paley–Wiener theorem Paley–Wiener theorem for the Fourier transform on Lie groups ⓘ linked to: Paley–Wiener theorem for real reductive groups Paley–Wiener theorem for the Fourier transform on R^n ⓘ linked to: Paley–Wiener theorem Paley–Wiener theorem for the Fourier transform on locally compact abelian groups ⓘ linked to: Paley–Wiener theorem
holdsFor	compactly supported C-infinity functions ⓘ compactly supported distributions ⓘ
implies	Fourier transform of a compactly supported function extends to an entire function ⓘ Fourier transform of a compactly supported function has exponential type ⓘ growth bounds for Fourier transforms in terms of support radius ⓘ
namedAfter	Norbert Wiener ⓘ Raymond Paley ⓘ linked to: Raymond E. A. C. Paley
relatedTo	Fourier inversion theorem ⓘ Paley–Wiener space ⓘ Paley–Wiener–Schwartz theorem ⓘ linked to: Paley–Wiener theorem Plancherel theorem ⓘ linked to: Plancherel theorem for locally compact abelian groups band-limited functions ⓘ uncertainty principle ⓘ
relates	analytic continuation of Fourier transforms ⓘ growth properties of entire functions ⓘ support properties of functions ⓘ
timePeriod	20th century mathematics ⓘ
usedIn	control theory ⓘ partial differential equations ⓘ representation theory ⓘ signal processing ⓘ spectral theory ⓘ time–frequency analysis ⓘ

How these facts were elicited