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:


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]) (, 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


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.

Title Parseval equality
Canonical name ParsevalEquality
Date of creation 2013-03-22 13:57:10
Last modified on 2013-03-22 13:57:10
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 11
Author asteroid (17536)
Entry type Theorem
Classification msc 42B05
Synonym Parseval equation
Synonym Parseval identity
Synonym Lyapunov equation
Related topic BesselInequality
Related topic ValueOfTheRiemannZetaFunctionAtS2
Defines Parseval theorem