Parseval's Theorem/Formulation 1

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:

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


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.



Proof


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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 {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



Source of Name

This entry was named for Marc-Antoine Parseval.

This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.