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.
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.