Parseval's theorem

E624505

mathematical theorem result in Fourier analysis

Parseval's theorem is a fundamental result in Fourier analysis that equates the total energy of a function in the time (or spatial) domain with the total energy of its representation in the frequency domain.

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All labels observed (2)

Label	Occurrences
Parseval theorem	2
Parseval's theorem canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T6858929 — 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: Parseval's theorem
Context triple: [Wiener–Khinchin theorem, relatedTo, Parseval's theorem]

A. 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.
B. 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.
C. Fourier transform
The Fourier transform is a mathematical operation that decomposes a function or signal into its constituent frequencies, widely used in engineering, physics, and signal processing.
D. Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
E. convolution theorem
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.
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: Parseval's theorem
Target entity description: Parseval's theorem is a fundamental result in Fourier analysis that equates the total energy of a function in the time (or spatial) domain with the total energy of its representation in the frequency domain.

A. 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.
B. 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.
C. Fourier transform
The Fourier transform is a mathematical operation that decomposes a function or signal into its constituent frequencies, widely used in engineering, physics, and signal processing.
D. Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
E. convolution theorem
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.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in Fourier analysis ⓘ
appliesTo	L2(-π,π) ⓘ L2(R) ⓘ square-integrable functions ⓘ
assumes	existence of Fourier representation ⓘ square-integrability of the function ⓘ
category	theorems in functional analysis ⓘ theorems in harmonic analysis ⓘ theorems in real analysis ⓘ
coreIdea	conservation of L2 norm under Fourier transform ⓘ equality of energy in time and frequency domains ⓘ
field	Fourier analysis NERFINISHED ⓘ applied mathematics ⓘ functional analysis ⓘ harmonic analysis ⓘ signal processing ⓘ
generalizedBy	Plancherel's theorem NERFINISHED ⓘ
historicalPeriod	19th century mathematics ⓘ
holdsIn	Hilbert spaces with orthonormal basis ⓘ
implies	Fourier transform is an isometry on L2 NERFINISHED ⓘ
isSpecialCaseOf	Plancherel's theorem NERFINISHED ⓘ
mathematicalFormulation	integral of \|f(x)\|^2 equals integral of \|F(ω)\|^2 up to normalization ⓘ sum of squares of Fourier coefficients equals L2 norm squared of function ⓘ
namedAfter	Marc-Antoine Parseval NERFINISHED ⓘ
relatedTo	Bessel's inequality NERFINISHED ⓘ orthogonality of trigonometric system ⓘ unitary operators ⓘ
relatesConcept	Fourier series NERFINISHED ⓘ Fourier transform NERFINISHED ⓘ L2 space ⓘ energy of a function ⓘ frequency domain ⓘ orthonormal basis ⓘ time domain ⓘ
usedFor	computing mean-square error in approximations ⓘ energy conservation checks in numerical algorithms ⓘ proving convergence properties of Fourier series ⓘ
usedIn	communications engineering ⓘ filter design ⓘ image processing ⓘ quantum mechanics ⓘ signal energy computation ⓘ spectral analysis ⓘ vibration analysis ⓘ

How these facts were elicited

Referenced by (3)

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

Wiener–Khinchin theorem → relatedTo → Parseval's theorem ⓘ

Cauchy–Schwarz inequality → relatedTo → Parseval's theorem ⓘ

this entity surface form: Parseval theorem

Riesz–Fischer theorem → relatedTo → Parseval's theorem ⓘ