Paley–Wiener theorem
E898512
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Paley–Wiener theorem canonical | 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]
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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.
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Paley–Wiener theorem for the Fourier transform on Lie groups NERFINISHED ⓘ Paley–Wiener theorem for the Fourier transform on R^n NERFINISHED ⓘ Paley–Wiener theorem for the Fourier transform on locally compact abelian groups NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Raymond Paley NERFINISHED ⓘ |
| relatedTo |
Fourier inversion theorem
NERFINISHED
ⓘ
Paley–Wiener space NERFINISHED ⓘ Paley–Wiener–Schwartz theorem NERFINISHED ⓘ Plancherel theorem NERFINISHED ⓘ 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Paley–Wiener theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.