convolution theorem
E556420
The convolution theorem is a fundamental result in Fourier analysis stating that convolution in one domain corresponds to pointwise multiplication in the Fourier-transformed domain (and vice versa), greatly simplifying the analysis of linear systems.
All labels observed (1)
| Label | Occurrences |
|---|---|
| convolution theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5935322 — 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: convolution theorem Context triple: [Fourier optics, uses, convolution theorem]
-
A.
Fourier inversion theorem
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.
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B.
FFT
FFT is the ICAO airline designator used in aviation to identify Frontier Airlines in flight plans and air traffic control communications.
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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.
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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E.
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.
- 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: convolution theorem Target entity description: The convolution theorem is a fundamental result in Fourier analysis stating that convolution in one domain corresponds to pointwise multiplication in the Fourier-transformed domain (and vice versa), greatly simplifying the analysis of linear systems.
-
A.
Fourier inversion theorem
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.
-
B.
FFT
FFT is the ICAO airline designator used in aviation to identify Frontier Airlines in flight plans and air traffic control communications.
-
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.
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.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in Fourier analysis ⓘ |
| appliesTo |
continuous-time signals
ⓘ
discrete-time signals ⓘ functions in L1 ⓘ functions in L2 ⓘ |
| category |
theorems in analysis
ⓘ
theorems in functional analysis ⓘ |
| dependsOn |
integral representation of the Fourier transform
ⓘ
linearity of the Fourier transform ⓘ |
| domain |
frequency domain
ⓘ
time domain ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ signal processing ⓘ systems theory ⓘ |
| generalizationOf |
similar properties for Laplace transform
ⓘ
similar properties for z-transform ⓘ |
| hasConsequence |
frequency response of an LTI system is the Fourier transform of its impulse response
ⓘ
output spectrum of an LTI system equals input spectrum times system frequency response ⓘ |
| hasFormulation |
Discrete-time version uses the discrete-time Fourier transform or DFT with circular convolution
ⓘ
F{f·g}(ω)=(1/2π)(F{f}∗F{g})(ω) for a common continuous-time convention ⓘ F{f∗g}(ω)=F{f}(ω)·F{g}(ω) ⓘ |
| hasVariant |
convolution theorem for distributions
ⓘ
convolution theorem for multidimensional Fourier transforms ⓘ convolution theorem for tempered distributions ⓘ |
| implies |
convolution in time domain corresponds to multiplication in frequency domain
ⓘ
multiplication in time domain corresponds to convolution in frequency domain ⓘ |
| relatesConcept |
Fourier transform
NERFINISHED
ⓘ
convolution ⓘ linear time-invariant systems ⓘ pointwise multiplication ⓘ |
| requires | existence of Fourier transforms of the functions involved ⓘ |
| statement |
The Fourier transform of a convolution is the pointwise product of the Fourier transforms.
ⓘ
The Fourier transform of a product is the convolution of the Fourier transforms, up to normalization factors. ⓘ |
| usedFor |
analysis of linear systems
ⓘ
computing convolutions efficiently via FFT ⓘ deconvolution ⓘ filter design ⓘ image processing ⓘ solving linear difference equations ⓘ solving linear differential equations ⓘ spectral analysis ⓘ system identification ⓘ |
| usedIn |
acoustics
ⓘ
communications engineering NERFINISHED ⓘ control engineering ⓘ optics ⓘ quantum mechanics NERFINISHED ⓘ |
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: convolution theorem Description of subject: The convolution theorem is a fundamental result in Fourier analysis stating that convolution in one domain corresponds to pointwise multiplication in the Fourier-transformed domain (and vice versa), greatly simplifying the analysis of linear systems.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.