Parseval's relation

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The name Parseval refers to mathematician Marc-Antoine Parseval (April 27, 1755 – August 16, 1836).


Statement of Parseval's relation

For functions and such that the Fourier transform and exist

Here and are the respective complex conjugates of and respectively.

Proof

We write the respective Fourier transform representations of and



and the complex conjugate of the second espression is


Forming the inner product

Rearranging the order of the integrations, we may write

Recognizing that the factor in is the Fourier representation of the Dirac delta function we may perform the integration via the sifting property of the delta function to yield Parseval's relation

When and are the same function,

where indicate that the modulus of a complex valued function is being squared and integrated.