From 8f675ddfda97b9298fe0e3503ce533d2c2bcd746 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Tue, 26 Apr 2022 00:19:09 +0200
Subject: Minor corrections and rewording

---
 notes/FourierOnS2.tex | 12 +++++++-----
 1 file changed, 7 insertions(+), 5 deletions(-)

(limited to 'notes/FourierOnS2.tex')

diff --git a/notes/FourierOnS2.tex b/notes/FourierOnS2.tex
index aeae7ef..2dc30c1 100644
--- a/notes/FourierOnS2.tex
+++ b/notes/FourierOnS2.tex
@@ -61,7 +61,7 @@ With this construction, now we just need a suitable set of basis function for th
 where \(m, n \in \mathbb{Z}\), be our basis functions in the space of ``nice'' functions from \(\mathbb{R}^2\) to \(\mathbb{C}\). Like in the one dimensional Fourier analysis, we can now define the Fourier coefficients.
 
 \begin{definition}[Fourier coefficients]
-  Let \(f : \mathbb{R}^2 \to \mathbb{C}\) be a ``nice'' function. The numbers
+  Let \(f(\mu, \nu) \in C(\mathbb{R}^2/\mathbb{Z}^2; \mathbb{C})\). The numbers
   \[
     c_{m, n} = \langle f, B_{m, n} \rangle
              = \iint_{[0, 1]^2} f(\mu, \nu) e^{-i2\pi (m\mu + n\nu)} d\mu d\nu,
@@ -69,12 +69,13 @@ where \(m, n \in \mathbb{Z}\), be our basis functions in the space of ``nice'' f
   are called the Fourier coefficients of \(f\).
 \end{definition}
 
-And finally by the Fourier theorem we can reconstruct the original function using an unoriginally named Fourier series:
+And finally by the Fourier theorem we can reconstruct the original function using a Fourier series:
 \[
   f(\mu, \nu) =
     \sum_{m\in\mathbb{Z}} \sum_{n\in\mathbb{Z}} c_{m, n} B_{m, n}(\mu, \nu).
 \]
 
+% TODO: keep or remove? <=> discuss fourier theorem?
 % \begin{definition}[\(L^2\) norm]
 %   \[
 %     \|f\|_2  = \sqrt{\langle f, f \rangle}
@@ -115,8 +116,9 @@ Notice that the partial derivatives have been simplified to normal derivatives.
 \]
 We let \(w\) be the separation constant, and we see that this results in two almost identical problems of the form
 \[
-  \frac{d^2 X}{d\chi^2} = \pm w X(\chi).
+  \frac{d^2 X}{d\xi^2} = \pm w X(\xi).
 \]
+% TODO: prove / discuss that w (=m or n) must be integers because of the periodicity of f
 The solutions to this elementary ODE are of course complex exponentials, the same we used to build the Fourier theory. This is not a coincidence, in fact quite the opposite: the basis functions of the Fourier decomposition were chosen such that the Laplacian operator is easy in the frequency domain. In other words, such that the expression
 \[
   \langle \nabla^2 f, B_{m, n} \rangle
@@ -124,14 +126,14 @@ The solutions to this elementary ODE are of course complex exponentials, the sam
 is easy to compute. This is shown in the next lemma.
 
 \begin{lemma}
-  Let \(f \in C(\mathbb{R}^2/\mathbb{Z}^2)\), then
+  Let \(f \in C(\mathbb{R}^2/\mathbb{Z}^2; \mathbb{C})\), then
   \[
     \langle \nabla^2 f, B_{m, n} \rangle
     = (2\pi i)^2 \left( m^2 + n^2 \right) \langle f, B_{m, n} \rangle.
   \]
 \end{lemma}
 \begin{proof}
-To begin this proof, we first expand the left side of the statement:
+To start, we first expand the left side of the statement:
 \begin{gather} 
   \nonumber
   \langle \nabla^2 f, B_{m, n} \rangle
-- 
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