From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/parsevals-theorem/index.md | 82 ++++++++++++++++++++++++++ 1 file changed, 82 insertions(+) create mode 100644 source/know/concept/parsevals-theorem/index.md (limited to 'source/know/concept/parsevals-theorem') diff --git a/source/know/concept/parsevals-theorem/index.md b/source/know/concept/parsevals-theorem/index.md new file mode 100644 index 0000000..43d3717 --- /dev/null +++ b/source/know/concept/parsevals-theorem/index.md @@ -0,0 +1,82 @@ +--- +title: "Parseval's theorem" +date: 2021-02-22 +categories: +- Mathematics +- Physics +layout: "concept" +--- + +**Parseval's theorem** is a relation between the inner product of two functions $f(x)$ and $g(x)$, +and the inner product of their [Fourier transforms](/know/concept/fourier-transform/) +$\tilde{f}(k)$ and $\tilde{g}(k)$. +There are two equivalent ways of stating it, +where $A$, $B$, and $s$ are constants from the FT's definition: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \Inprod{f(x)}{g(x)} &= \frac{2 \pi B^2}{|s|} \inprod{\tilde{f}(k)}{\tilde{g}(k)} + \\ + \inprod{\tilde{f}(k)}{\tilde{g}(k)} &= \frac{2 \pi A^2}{|s|} \Inprod{f(x)}{g(x)} + \end{aligned} + } +\end{aligned}$$ + +
+ + + +
+ +For this reason, physicists like to define the Fourier transform +with $A\!=\!B\!=\!1 / \sqrt{2\pi}$ and $|s|\!=\!1$, because then it nicely +conserves the functions' normalization. + + + +## References +1. O. Bang, + *Applied mathematics for physicists: lecture notes*, 2019, + unpublished. -- cgit v1.2.3