Parseval's Theorem

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Theorem

Formulation 1

Let $f$ be a real function which is square-integrable over the interval $\left[{-\pi \,.\,.\, \pi}\right]$.

Let $f$ be expressed by the Fourier series:

$\displaystyle f \left({x}\right) \cong \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \left({a_n \cos \left({n x}\right) + b_n \sin \left({n x}\right)}\right)$


Then:

$\displaystyle \frac 1 \pi \int_{-\pi}^\pi \left\lvert{f^2 \left({x}\right)}\right\rvert \, \mathrm d x = \frac { {a_0}^2} 2 + \sum_{n \mathop = 1}^\infty \left({ {a_n}^2 + {b_n}^2}\right)$


Formulation 2

Let $f$ be a real function which is square-integrable over the interval $\left[{-\pi \,.\,.\, \pi}\right]$.

Let $f$ be expressed by the Fourier series:

$\displaystyle f \left({x}\right) = \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$

where:

$\displaystyle c_n = \frac 1 {2 \pi} \int_{-\pi}^\pi f \left({t}\right) e^{-i n t} \, \mathrm d t$


Then:

$\displaystyle \frac 1 {2 \pi} \int_{-\pi}^\pi \left\lvert{f \left({x}\right)}\right\rvert^2 \, \mathrm d x = \sum_{n \mathop = -\infty}^\infty \left\lvert{c_n}\right\rvert^2$


Source of Name

This entry was named for Marc-Antoine Parseval.