Parseval's Theorem

Theorem


This article is complete as far as it goes, but it could do with expansion.
In particular: Make the interval general: $\closedint \alpha {\alpha + 2 \pi}$ for arbitrary $\alpha$, in order for consistency with the work on Fourier analysis as it is currently in progress.
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Formulation 1

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

Let $f$ be expressed by the Fourier series:

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


This article, or a section of it, needs explaining.
In particular: What does $\sim$ mean?
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Then:

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


This article, or a section of it, needs explaining.
In particular: Is this not $\size {\map f x}^2$? I know these are the same but the latter is more common.
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Formulation 2

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

Let $f$ be expressed by the Fourier series:

$\map f x = \ds \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$

where:

$c_n = \ds \frac 1 {2 \pi} \int_{-\pi}^\pi \map f t e^{-i n t} \rd t$


Then:

$\ds \frac 1 {2 \pi} \int_{-\pi}^\pi \size {\map f x}^2 \rd x = \sum_{n \mathop = -\infty}^\infty \size {c_n}^2$


Also known as

is also known as Parseval's Identity.


Source of Name

This entry was named for Marc-Antoine Parseval.

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