# 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 ${\displaystyle f(t)}$ and ${\displaystyle g(t)}$ such that the Fourier transform ${\displaystyle F(\omega )}$ and ${\displaystyle G(\omega )}$ exist

${\displaystyle \int _{-\infty }^{\infty }f(t){\overline {g(t)}}\;dt={\frac {1}{2\pi }}F(\omega ){\overline {G(\omega )}}\;d\omega .}$

Here ${\displaystyle {\overline {G(\omega )}}}$ and ${\displaystyle {\overline {g(t)}}}$ are the respective complex conjugates of ${\displaystyle G(\omega )}$ and ${\displaystyle g(t),}$ respectively.

### Proof

We write the respective Fourier transform representations of ${\displaystyle f(t)}$ and ${\displaystyle g(t)}$

${\displaystyle f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )e^{-i\omega t}\;d\omega }$

${\displaystyle g(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }g(\Omega )e^{-i\Omega t}\;d\Omega }$

and the complex conjugate of the second espression is

${\displaystyle {\overline {g(t)}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\overline {g(\Omega )}}e^{i\Omega t}\;d\Omega }$

Forming the inner product

${\displaystyle \int _{-\infty }^{\infty }f(t){\overline {g(t)}}\;dt={\frac {1}{(2\pi )^{2}}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\;d\omega \;\int _{-\infty }^{\infty }\;d\Omega \;F(\omega ){\overline {G(\Omega )}}e^{-i(\omega -\Omega )t}\;dt.}$

Rearranging the order of the integrations, we may write

${\displaystyle \int _{-\infty }^{\infty }f(t){\overline {g(t)}}\;dt={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\;d\omega \;\int _{-\infty }^{\infty }\;d\Omega \;F(\omega ){\overline {G(\Omega )}}\left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{-i(\omega -\Omega )t}\;dt.\right]}$

Recognizing that the factor in ${\displaystyle [...]}$ is the Fourier representation of the Dirac delta function ${\displaystyle \delta (\Omega -\omega )}$ we may perform the ${\displaystyle \Omega }$ integration via the sifting property of the delta function to yield Parseval's relation

${\displaystyle \int _{-\infty }^{\infty }f(t){\overline {g(t)}}\;dt={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega ){\overline {G(\omega )}}\;d\omega .}$

When ${\displaystyle f(t)}$ and ${\displaystyle g(t)}$ are the same function,

${\displaystyle \int _{-\infty }^{\infty }|f(t)|^{2}\;dt={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|F(\omega )|^{2}\;d\omega ,}$

where ${\displaystyle |...|}$ indicate that the modulus of a complex valued function is being squared and integrated.