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单词 ParsevalEquality
释义

Parseval equality


0.1 Parseval’s Equality

Theorem. – If {ej:jJ} is an orthonormal basis of an Hilbert spaceMathworldPlanetmath H, then for every xH the following equality holds:

x2=jJ|x,ej|2.

The above theorem is a more sophisticated form of Bessel’s inequalityMathworldPlanetmath (where the inequality is in fact an equality). The difference is that for Bessel’s inequality it is only required that the set {ej:jJ} is an orthonormal set, not necessarily an orthonormal basis.

0.2 Parseval’s Theorem

Applying Parseval’s equality on the Hilbert space L2([-π,π]) (http://planetmath.org/LpSpace), of square integrable functionsMathworldPlanetmath on the interval [-π,π], with the orthonormal basis consisting of trigonometric functionsDlmfMathworldPlanetmath, we obtain

Theorem (Parseval’s theorem). – Let f be a Riemann square integrable function from [-π,π] to .  The following equality holds

1π-ππf2(x)𝑑x=(a0f)22+k=1[(akf)2+(bkf)2],

where a0f, akf, bkf are the Fourier coefficients of the function f.

The function f can be a Lebesgue-integrable function, if we use the Lebesgue integralMathworldPlanetmath in place of the Riemann integral.

TitleParseval equality
Canonical nameParsevalEquality
Date of creation2013-03-22 13:57:10
Last modified on2013-03-22 13:57:10
Ownerasteroid (17536)
Last modified byasteroid (17536)
Numerical id11
Authorasteroid (17536)
Entry typeTheorem
Classificationmsc 42B05
SynonymParseval equation
SynonymParseval identity
SynonymLyapunov equation
Related topicBesselInequality
Related topicValueOfTheRiemannZetaFunctionAtS2
DefinesParseval theorem
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