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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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

Parseval's Theorem is also known as Parseval's Identity.

Source of Name

This entry was named for Marc-Antoine Parseval.