Parseval's Theorem/Formulation 1

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Theorem

Let $f$ be a real function which is square-integrable over the interval $\openint {-\pi} \pi$.

Let $f$ be expressed by the Fourier series:

$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$


Then:

$\displaystyle \frac 1 \pi \int_{-\pi}^\pi \size {\map {f^2} x} \rd x = \frac { {a_0}^2} 2 + \sum_{n \mathop = 1}^\infty \paren { {a_n}^2 + {b_n}^2}$


Proof


Source of Name

This entry was named for Marc-Antoine Parseval.