diff options
| -rw-r--r-- | buch/papers/kugel/applications.tex | 7 | ||||
| -rw-r--r-- | buch/papers/kugel/preliminaries.tex | 6 | ||||
| -rw-r--r-- | buch/papers/kugel/spherical-harmonics.tex | 81 |
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} |