summaryrefslogtreecommitdiffstats
path: root/notes
diff options
context:
space:
mode:
authorNao Pross <np@0hm.ch>2022-04-26 00:19:09 +0200
committerNao Pross <np@0hm.ch>2022-04-26 00:21:14 +0200
commit8f675ddfda97b9298fe0e3503ce533d2c2bcd746 (patch)
tree5d6f2123b721f8ebc9b3f2d4f965c3cc6d3eecff /notes
parentFix mistake in notes on Fourer on a flat surface (diff)
downloadFourierOnS2-8f675ddfda97b9298fe0e3503ce533d2c2bcd746.tar.gz
FourierOnS2-8f675ddfda97b9298fe0e3503ce533d2c2bcd746.zip
Minor corrections and rewording
Diffstat (limited to 'notes')
-rw-r--r--notes/FourierOnS2.tex12
-rw-r--r--notes/build/FourierOnS2.pdfbin45497 -> 44823 bytes
-rw-r--r--notes/tex/docstyle.sty2
3 files changed, 8 insertions, 6 deletions
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
diff --git a/notes/build/FourierOnS2.pdf b/notes/build/FourierOnS2.pdf
index 7316cc2..0201255 100644
--- a/notes/build/FourierOnS2.pdf
+++ b/notes/build/FourierOnS2.pdf
Binary files differ
diff --git a/notes/tex/docstyle.sty b/notes/tex/docstyle.sty
index f813c99..a094238 100644
--- a/notes/tex/docstyle.sty
+++ b/notes/tex/docstyle.sty
@@ -3,7 +3,7 @@
%% Margins
\RequirePackage{geometry}
-\newgeometry{a4paper, margin=2.2cm, top=3cm, bottom=3cm}
+\newgeometry{a4paper, margin=2cm, top=3cm, bottom=3cm}
%% Headers and footers
\RequirePackage{fancyhdr}