Fourier inversion theorem
E259775
The Fourier inversion theorem is a fundamental result in harmonic analysis that guarantees, under suitable conditions, that a function can be exactly reconstructed from its Fourier transform.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Fourier inversion theorem canonical | 1 |
| Fourier transform is invertible on appropriate function spaces | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364629 — 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: Fourier inversion theorem Context triple: [Riemann–Lebesgue lemma, relatedTo, Fourier inversion theorem]
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A.
Fourier analysis
Fourier analysis is a mathematical method for decomposing functions or signals into sums of sinusoidal components, widely used in fields such as signal processing, physics, and engineering.
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B.
Fourier
Fourier is a French surname most famously associated with Jean-Baptiste Joseph Fourier, the mathematician and physicist known for developing Fourier analysis and Fourier series.
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C.
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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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.
The Fourier Integral and Certain of Its Applications
The Fourier Integral and Certain of Its Applications is a foundational mathematical work by Norbert Wiener that develops and applies Fourier analysis to problems in harmonic analysis and related areas.
- 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: Fourier inversion theorem Target entity description: The Fourier inversion theorem is a fundamental result in harmonic analysis that guarantees, under suitable conditions, that a function can be exactly reconstructed from its Fourier transform.
-
A.
Fourier analysis
Fourier analysis is a mathematical method for decomposing functions or signals into sums of sinusoidal components, widely used in fields such as signal processing, physics, and engineering.
-
B.
Fourier
Fourier is a French surname most famously associated with Jean-Baptiste Joseph Fourier, the mathematician and physicist known for developing Fourier analysis and Fourier series.
-
C.
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.
-
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.
The Fourier Integral and Certain of Its Applications
The Fourier Integral and Certain of Its Applications is a foundational mathematical work by Norbert Wiener that develops and applies Fourier analysis to problems in harmonic analysis and related areas.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in harmonic analysis ⓘ |
| appliesTo |
functions on Euclidean space
ⓘ
functions on the real line ⓘ integrable functions ⓘ square-integrable functions ⓘ |
| concerns |
almost-everywhere convergence of inverse Fourier integrals
ⓘ
convergence in Lp norms for certain p ⓘ pointwise convergence of inverse Fourier integrals ⓘ |
| dealsWith |
Fourier transform
ⓘ
function reconstruction ⓘ integral transforms ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ |
| generalizes | inversion formulas for Fourier series ⓘ |
| guarantees | reconstruction of a function from its Fourier transform under suitable conditions ⓘ |
| hasAssumption | choice of a specific Fourier transform normalization convention ⓘ |
| hasConsequence |
equivalence between time-domain and frequency-domain descriptions of a function
ⓘ
uniqueness of Fourier transform representation under given conditions ⓘ |
| hasHistoricalContext | developed in the late 19th and early 20th centuries in analysis ⓘ |
| hasVersion |
L1 version
ⓘ
L2 version ⓘ Schwartz space version ⓘ tempered distributions version ⓘ |
| holdsIn |
Euclidean space
ⓘ
surface form:
Euclidean spaces Rn
|
| implies |
Fourier inversion theorem
self-linksurface differs
ⓘ
surface form:
Fourier transform is invertible on appropriate function spaces
equality of a function and its inverse Fourier transform in L2 sense under suitable hypotheses ⓘ original function can be recovered almost everywhere from its Fourier transform ⓘ |
| isFormulatedUsing |
Hilbert space methods
ⓘ
Lebesgue integration ⓘ measure theory ⓘ |
| isRelatedTo |
Fourier series
ⓘ
Fourier transform on L2 ⓘ Paley–Wiener theorem ⓘ Plancherel theorem for locally compact abelian groups ⓘ
surface form:
Plancherel theorem
Poisson summation formula ⓘ Riemann–Lebesgue lemma ⓘ distribution theory ⓘ |
| isUsedIn |
image processing
ⓘ
partial differential equations ⓘ quantum mechanics ⓘ signal processing ⓘ time–frequency analysis ⓘ |
| requires |
conditions on continuity or differentiability at points of reconstruction
ⓘ
conditions on decay at infinity ⓘ integrability conditions on the function ⓘ suitable regularity conditions on the function ⓘ |
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Fourier inversion theorem Description of subject: The Fourier inversion theorem is a fundamental result in harmonic analysis that guarantees, under suitable conditions, that a function can be exactly reconstructed from its Fourier transform.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Fourier transform is invertible on appropriate function spaces