From a39bb3b8aab1aeb4fceaedc54c756703819776c3 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 17 Dec 2022 18:19:26 +0100
Subject: Rewrite "Lagrange multiplier", various improvements

---
 source/know/concept/parsevals-theorem/index.md | 20 +++++++++++---------
 1 file changed, 11 insertions(+), 9 deletions(-)

(limited to 'source/know/concept/parsevals-theorem/index.md')

diff --git a/source/know/concept/parsevals-theorem/index.md b/source/know/concept/parsevals-theorem/index.md
index 377f3a1..41e8fed 100644
--- a/source/know/concept/parsevals-theorem/index.md
+++ b/source/know/concept/parsevals-theorem/index.md
@@ -26,20 +26,21 @@ $$\begin{aligned}
 
 
 {% include proof/start.html id="proof-fourier" -%}
-We insert the inverse FT into the defintion of the inner product:
+We insert the inverse FT into the definition of the inner product:
 
 $$\begin{aligned}
     \Inprod{f}{g}
     &= \int_{-\infty}^\infty \big( \hat{\mathcal{F}}^{-1}\{\tilde{f}(k)\}\big)^* \: \hat{\mathcal{F}}^{-1}\{\tilde{g}(k)\} \dd{x}
     \\
     &= B^2 \int
-    \Big( \int \tilde{f}^*(k_1) \exp(i s k_1 x) \dd{k_1} \Big)
-    \Big( \int \tilde{g}(k) \exp(- i s k x) \dd{k} \Big)
+    \Big( \int \tilde{f}^*(k') \: e^{i s k' x} \dd{k'} \Big)
+    \Big( \int \tilde{g}(k) \: e^{- i s k x} \dd{k} \Big)
     \dd{x}
     \\
-    &= 2 \pi B^2 \iint \tilde{f}^*(k_1) \tilde{g}(k) \Big( \frac{1}{2 \pi} \int_{-\infty}^\infty \exp(i s x (k_1 - k)) \dd{x} \Big) \dd{k_1} \dd{k}
+    &= 2 \pi B^2 \iint \tilde{f}^*(k') \: \tilde{g}(k) \Big( \frac{1}{2 \pi}
+    \int_{-\infty}^\infty e^{i s x (k' - k)} \dd{x} \Big) \dd{k'} \dd{k}
     \\
-    &= 2 \pi B^2 \iint \tilde{f}^*(k_1) \: \tilde{g}(k) \: \delta(s (k_1 - k)) \dd{k_1} \dd{k}
+    &= 2 \pi B^2 \iint \tilde{f}^*(k') \: \tilde{g}(k) \: \delta\big(s (k' \!-\! k)\big) \dd{k'} \dd{k}
     \\
     &= \frac{2 \pi B^2}{|s|} \int_{-\infty}^\infty \tilde{f}^*(k) \: \tilde{g}(k) \dd{k}
     = \frac{2 \pi B^2}{|s|} \inprod{\tilde{f}}{\tilde{g}}
@@ -54,13 +55,14 @@ $$\begin{aligned}
     &= \int_{-\infty}^\infty \big( \hat{\mathcal{F}}\{f(x)\}\big)^* \: \hat{\mathcal{F}}\{g(x)\} \dd{k}
     \\
     &= A^2 \int
-    \Big( \int f^*(x_1) \exp(- i s k x_1) \dd{x_1} \Big)
-    \Big( \int g(x) \exp(i s k x) \dd{x} \Big)
+    \Big( \int f^*(x') \: e^{- i s k x'} \dd{x'} \Big)
+    \Big( \int g(x) \: e^{i s k x} \dd{x} \Big)
     \dd{k}
     \\
-    &= 2 \pi A^2 \iint f^*(x_1) g(x) \Big( \frac{1}{2 \pi} \int_{-\infty}^\infty \exp(i s k (x_1 - x)) \dd{k} \Big) \dd{x_1} \dd{x}
+    &= 2 \pi A^2 \iint f^*(x') \: g(x) \Big( \frac{1}{2 \pi}
+    \int_{-\infty}^\infty e^{i s k (x - x')} \dd{k} \Big) \dd{x'} \dd{x}
     \\
-    &= 2 \pi A^2 \iint f^*(x_1) \: g(x) \: \delta(s (x_1 - x)) \dd{x_1} \dd{x}
+    &= 2 \pi A^2 \iint f^*(x') \: g(x) \: \delta\big(s (x \!-\! x')\big) \dd{x'} \dd{x}
     \\
     &= \frac{2 \pi A^2}{|s|} \int_{-\infty}^\infty f^*(x) \: g(x) \dd{x}
     = \frac{2 \pi A^2}{|s|} \Inprod{f}{g}
-- 
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