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authorNao Pross <np@0hm.ch>2022-08-30 22:42:54 +0200
committerNao Pross <np@0hm.ch>2022-08-30 22:42:54 +0200
commit1bdb803ced744bcfe7cf81c89a740fcbcf6bdc70 (patch)
treee2ce322601e737494626056741174ff5acf11e07 /buch
parentkugel: Minor changes and fix proofs (remove enumerate) (diff)
downloadSeminarSpezielleFunktionen-1bdb803ced744bcfe7cf81c89a740fcbcf6bdc70.tar.gz
SeminarSpezielleFunktionen-1bdb803ced744bcfe7cf81c89a740fcbcf6bdc70.zip
kugel: Minor corrections
Diffstat (limited to '')
-rw-r--r--buch/papers/kugel/applications.tex7
-rw-r--r--buch/papers/kugel/preliminaries.tex6
-rw-r--r--buch/papers/kugel/spherical-harmonics.tex81
3 files changed, 59 insertions, 35 deletions
diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index 15a57d5..10bf153 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -32,7 +32,6 @@ at their location (figure \ref{kugel:fig:eeg-electrodes}). To see how this will
relate to the spherical harmonics, we will first quickly recap a bit of physics,
electrodynamics to be precise.
-
\subsubsection{Electrodynamics}
In section \ref{kugel:sec:construction:eigenvalue} we have shown that the
@@ -88,7 +87,7 @@ using the spherical harmonics.
To solve this new interpolation problem, we will start with a blatantly
engineering assumption: the human head is a sphere of radius $R$, with the value
-of $R$ begin the average radius of a human head (which is around 11 cm). So, we
+of $R$ being the average radius of a human head (which is around 11 cm). So, we
will assume that the potential distribution on the head can be written as a
finite linear combination of spherical harmonics:
\begin{equation*}
@@ -110,7 +109,7 @@ minimization problem:
\sin \vartheta \, d\vartheta d\varphi,
\label{kugel:eqn:eeg-min} \\
&\text{under the constraints} \quad V(p_j) = \phi_j
- \quad \text{ for } \quad 1 < j < M.
+ \quad \text{ for } \quad 1 \leq j \leq M.
\label{kugel:eqn:eeg-min-constraints}
\end{align}
\end{subequations}
@@ -203,4 +202,4 @@ quite unwieldy.
\subsection{Measuring Gravitational Fields}
-\subsection{Quantisation of Angular Momentum}
+% \subsection{Quantisation of Angular Momentum}
diff --git a/buch/papers/kugel/preliminaries.tex b/buch/papers/kugel/preliminaries.tex
index 1fa78d7..c4c5cae 100644
--- a/buch/papers/kugel/preliminaries.tex
+++ b/buch/papers/kugel/preliminaries.tex
@@ -288,7 +288,7 @@ way that from now on we will not have to worry about the details of
convergence.
\begin{lemma}
- \label{kugel:lemma:exp-1d}
+ \label{kugel:thm:exp-1d}
The set of functions \(E_n(x) = e^{i2\pi nx}\) on the interval
\([0; 1)\) with \(n \in \mathbb{Z} \) are orthonormal.
\end{lemma}
@@ -318,7 +318,7 @@ convergence.
\end{definition}
\begin{theorem}[Fourier Theorem]
- \label{fourier-theorem-1D}
+ \label{kugel:thm:fourier-theorem}
\begin{equation*}
\lim_{N \to \infty} \left \|
f(x) - \sum_{n = -N}^N \hat{f}(n) E_n(x)
@@ -331,7 +331,7 @@ convergence.
on the square \([0; 1)^2\) with \(m, n \in \mathbb{Z} \) are orthonormal.
\end{lemma}
\begin{proof}
- The proof is almost identical to lemma \ref{kugel:lemma:exp-1d}, with the
+ The proof is almost identical to lemma \ref{kugel:thm:exp-1d}, with the
only difference that the inner product is given by
\[
\langle E_{m,n}, E_{m', n'} \rangle
diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 5f5d59e..68b7eda 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -740,7 +740,7 @@ part, where we will derive the recursion equations for
$Y^m_n(\vartheta,\varphi)$, we will basically reuse the ones presented above.
Maybe it is worth mentioning at least one use case for these relations: In some
software implementations (that include lighting computations in computer
-graphics, antenna modelling software, 3-D modelling in medical applications,
+graphics, antenna modelling software, 3D modelling in medical applications,
etc.) they are widely used, as they lead to better numerical accuracy and
computational cost lower by a factor of six \cite{davari_new_2013}.
@@ -990,14 +990,21 @@ functions. We can say that the theory we are about to show can be applied on
all twice differentiable complex valued functions, to be more concise: complex
valued $L^2$ functions $S^2 \to \mathbb{C}$.
-All these jargons are not really necessary for the practical applications of us mere mortals, namely physicists and engineers.
-From now on we will therefore assume that the functions we will dealing with fulfill these ``minor'' conditions.
+All this jargon is not really necessary for the practical applications of us
+mere mortals, namely physicists and engineers. From now on we will therefore
+assume that the functions we will dealing with fulfill these ``minor''
+conditions.
-The insiders could turn up their nose, but we don't want to dwell too much on the concept of Hilbert space, convergence, metric, well-behaved functions etc.
-We simply think that this rigorousness could be at the expense of the possibility to appreciate the beauty and elegance of this theory.
-Furthermore, the risk of writing 300+ pages to prove that $1+1=2$\cite{principia-mathematica} is just around the corner (we apologize in advance to Mr. Whitehead and Mr. Russel for using their effort with a negative connotation).
+The insiders could turn up their nose, but we don't want to dwell too much on
+the concept of Hilbert space, convergence, metric, well-behaved functions etc.
+We simply think that this rigorousness could be at the expense of the
+possibility to appreciate the beauty and elegance of this theory. Furthermore,
+the risk of writing 300+ pages to prove that $1+1=2$\cite{principia-mathematica}
+is just around the corner (we apologize in advance to Mr. Whitehead and Mr.
+Russel for using their effort with a negative connotation).
-Despite all, if you desire having definitions a bit more rigorous (as rigorous as two engineers can be), you could take a look at the chapter \ref{}.
+Despite all, if you desire having definitions a bit more rigorous (as rigorous
+as two engineers can be), you could take a look at the chapter \ref{}.
\subsection{Spherical Harmonics Series}
@@ -1005,54 +1012,72 @@ To talk about a \emph{series expansion} we first need a series, so we shall
build one using the spherical harmonics.
\begin{definition}[Spherical harmonic series]
- \label{kugel:definition:spherical-harmonics-series}
+ \label{kugel:def:spherical-harmonics-series}
\begin{equation}
f(\vartheta, \varphi)
= \sum_{n=0}^\infty \sum_{m =-n}^n
- c_{m,n} Y^m_n(\vartheta, \varphi). \label{kugel:definition:spherical-harmonics-series}
+ c_{m,n} Y^m_n(\vartheta, \varphi).
+ \label{kugel:eqn:spherical-harmonics-series}
\end{equation}
\end{definition}
-With this definition we are basically saying that any function defined on the spherical surface can be represented as a linear combination of spherical harmonics.
-Does eq.\eqref{kugel:definition:spherical-harmonics-series} sound familiar? Well that is prefectly normal, since this is analog to the classical Fourier theory.
-In the latter is stated that ``any'' $T$-periodic function $f(x)$, on any interval $[x_0-T/2,x_0+T/2]$, can be represented as a linear combination of complex exponentials. More compactly:
+With this definition we are basically saying that any function defined on the
+spherical surface can be represented as a linear combination of spherical
+harmonics. Does equation \eqref{kugel:definition:spherical-harmonics-series}
+sound familiar? Well that is perfectly normal, since this is analog to the
+classical Fourier theory. In the latter is stated that ``any'' $T$-periodic
+function $f(x)$, on any interval $[x_0-T/2,x_0+T/2]$, can be represented as a
+linear combination of complex exponentials. More compactly:
\begin{equation*}
- f(x) = \sum_{n \in \mathbb{Z}} c_n e^{i \omega_0 x}, \quad \omega_0=\frac{2\pi}{T}
+ f(x) = \sum_{n \in \mathbb{Z}} c_n e^{i \omega_0 x},
+ \quad \omega_0=\frac{2\pi}{T}
\end{equation*}
-In the case of definition \ref{kugel:definition:spherical-harmonics-series} the kernels, instead of $e^{i\omega_0x}$, have become $Y^m_n$. In addition, the sum is now over the two indices $m$ and $n$.
+In the case of definition \ref{kugel:def:spherical-harmonics-series} the
+kernels, instead of $e^{i\omega_0x}$, have become $Y^m_n$. In addition, the sum
+is now over the two indices $m$ and $n$.
\begin{lemma}[Spherical harmonic coefficients]
- \label{kugel:lemma:spherical-harmonic-coefficient}
- \begin{align*}
+ \label{kugel:thm:spherical-harmonic-coefficient}
+ \begin{equation*}
c_{m,n}
- &= \langle f, Y^m_n \rangle_{\partial S} \\
- &= \int_0^\pi \int_0^{2\pi} f(\vartheta,\varphi) \overline{Y^m_n(\vartheta,\varphi)} \sin\vartheta \,d\varphi\,d\vartheta
- \end{align*}
+ = \langle f, Y^m_n \rangle_{\partial S}
+ = \int_0^\pi \int_0^{2\pi}
+ f(\vartheta,\varphi) \overline{Y^m_n(\vartheta,\varphi)}
+ \sin\vartheta \,d\varphi\,d\vartheta
+ \end{equation*}
+ \kugeltodo{Would be better as a definition? I don't get what is being proved.}
\end{lemma}
+
\begin{proof}
- To develop this proof we will take advantage of the orthogonality property of the Spherical Harmonics. We can start and finish by applying the inner product on both sides of eq.\eqref{kugel:definition:spherical-harmonics-series}:
+ To develop this proof we will take advantage of the orthogonality property of
+ the Spherical Harmonics. We can start and finish by applying the inner product
+ on both sides of \eqref{kugel:eqn:spherical-harmonics-series}:
\begin{align*}
\langle f, Y^{m}_{n} \rangle_{\partial S}
&= \left\langle \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'}
- c_{m',n'} Y^{m'}_{n'}(\vartheta, \varphi) \right\rangle_{\partial S} \\
- &= \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'}
+ c_{m',n'} Y^{m'}_{n'} \right\rangle_{\partial S} \\
+ &= \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'}
\langle c_{m',n'} Y^{m'}_{n'}, Y^{m}_{n} \rangle_{\partial S} \\
- &= \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'} c_{m',n'} \langle Y^{m'}_{n'}, Y^{m}_{n} \rangle_{\partial S} = c_{m,n}
+ &= \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'} c_{m',n'}
+ \langle Y^{m'}_{n'}, Y^{m}_{n} \rangle_{\partial S} = c_{m,n}
\end{align*}
- We omitted the $\vartheta, \varphi$ dependency to avoid overloading the notation.
\end{proof}
-Thanks to Lemma \ref{kugel:lemma:spherical-harmonic-coefficient} we can now calculate the series expansion defined in \ref{kugel:definition:spherical-harmonics-series}.
-It can be shown that, for the famous ``well-behaved functions'' $f(\vartheta, \varphi)$ mentioned before, theorem \ref{fourier-theorem-spherical-surface} is true
+Thanks to Lemma \ref{kugel:thm:spherical-harmonic-coefficient} we can now
+calculate the series expansion defined in
+\ref{kugel:def:spherical-harmonics-series}. It can be shown that, for the famous
+``well-behaved functions'' $f(\vartheta, \varphi)$ mentioned before, the
+following theorem is true. The connection to Theorem
+\ref{kugel:thm:fourier-theorem} is pretty obvious.
+
\begin{theorem}[Fourier Theorem on $\partial S$]
\label{fourier-theorem-spherical-surface}
\begin{equation*}
- \lim_{N \to \infty}
+ \lim_{N \to \infty}
\int_0^\pi \int_0^{2\pi} \left\| f(\vartheta,\varphi) - \sum_{n=0}^N\sum_{m=-n}^n c_{m,n} Y^m_n(\vartheta,\varphi)
\right\|_2 \sin\vartheta \,d\varphi\,d\vartheta = 0
\end{equation*}
\end{theorem}
-The connection to Theorem \ref{fourier-theorem-1D} is pretty obvious.
\subsection{Spectrum}