Parseval's Theorem

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Theorem

Let $f$ be a function square-integrable over $\left[-\pi \ldots \pi\right]$ given by the Fourier series,

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

Then,

$\displaystyle \frac 1 \pi \int_{-\pi}^\pi \left|f^2(x)\right|\mathrm dx = \frac {a_0^2} 2 + \sum_{n=1}^\infty \left(a^2_n + b^2_n\right)$

Proof


Source of Name

This entry was named for Marc-Antoine Parseval.