From 6e70f28ccbd5afc1506f71f013278a9d157ef03a Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 27 Oct 2022 20:40:09 +0200
Subject: Optimize last images, add proof template, improve CSS
---
source/know/concept/fourier-transform/index.md | 23 ++++++++---------------
1 file changed, 8 insertions(+), 15 deletions(-)
(limited to 'source/know/concept/fourier-transform/index.md')
diff --git a/source/know/concept/fourier-transform/index.md b/source/know/concept/fourier-transform/index.md
index 0bc849b..c86d997 100644
--- a/source/know/concept/fourier-transform/index.md
+++ b/source/know/concept/fourier-transform/index.md
@@ -67,6 +67,7 @@ on whether the analysis is for forward ($$s > 0$$) or backward-propagating
($$s < 0$$) waves.
+
## Derivatives
The FT of a derivative has a very useful property.
@@ -113,6 +114,7 @@ $$\begin{aligned}
\end{aligned}$$
+
## Multiple dimensions
The Fourier transform is straightforward to generalize to $$N$$ dimensions.
@@ -150,11 +152,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% include proof/start.html id="proof-constants-ndim" -%}
The inverse FT of the forward FT of $$f(\vb{x})$$ must be equal to $$f(\vb{x})$$ again, so:
$$\begin{aligned}
@@ -180,9 +179,8 @@ $$\begin{aligned}
&= \frac{(2 \pi)^N A B}{|s|^N} \int f(\vb{x}') \: \delta(\vb{x}' - \vb{x}) \ddn{N}{\vb{x}'}
= \frac{(2 \pi)^N A B}{|s|^N} f(\vb{x})
\end{aligned}$$
+{% include proof/end.html id="proof-constants-ndim" %}
-
-
Differentiation is more complicated for $$N > 1$$,
but the FT is still useful,
@@ -197,11 +195,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% include proof/start.html id="proof-laplacian" -%}
We insert $$\nabla^2 f$$ into the FT,
decompose the exponential and the Laplacian,
and then integrate by parts (limits $$\pm \infty$$ omitted):
@@ -236,9 +231,7 @@ $$\begin{aligned}
&= - A s^2 \sum_{n = 1}^N k_n^2 \int f \exp(i s \vb{k} \cdot \vb{x}) \ddn{N}{\vb{x}}
= - s^2 \sum_{n = 1}^N k_n^2 \tilde{f}
\end{aligned}$$
-
-
-
+{% include proof/end.html id="proof-laplacian" %}
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