diff options
Diffstat (limited to 'content/know/concept/parsevals-theorem/index.pdc')
| -rw-r--r-- | content/know/concept/parsevals-theorem/index.pdc | 15 |
1 files changed, 7 insertions, 8 deletions
diff --git a/content/know/concept/parsevals-theorem/index.pdc b/content/know/concept/parsevals-theorem/index.pdc index 8f653f8..ae34bda 100644 --- a/content/know/concept/parsevals-theorem/index.pdc +++ b/content/know/concept/parsevals-theorem/index.pdc @@ -13,8 +13,8 @@ markup: pandoc # Parseval's theorem -**Parseval's theorem** relates the inner product of two functions $f(x)$ and $g(x)$ to the -inner product of their [Fourier transforms](/know/concept/fourier-transform/) +**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 Fourier transform's definition: @@ -29,12 +29,11 @@ $$\begin{aligned} } \end{aligned}$$ -For this reason, physicists like to define their Fourier transform -with $A = B = 1 / \sqrt{2\pi}$ and $|s| = 1$, because then the FT nicely -conserves the total probability (quantum mechanics) or the total energy -(optics). +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. -To prove this, we insert the inverse FT into the inner product +To prove the theorem, we insert the inverse FT into the inner product definition: $$\begin{aligned} @@ -55,7 +54,7 @@ $$\begin{aligned} \end{aligned}$$ Where $\delta(k)$ is the [Dirac delta function](/know/concept/dirac-delta-function/). -Note that we can just as well do it in the opposite direction, +Note that we can equally well do the proof in the opposite direction, which yields an equivalent result: $$\begin{aligned} |