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authorErik Löffler <100943759+erik-loeffler@users.noreply.github.com>2022-08-23 15:03:32 +0200
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-rw-r--r--buch/papers/kugel/spherical-harmonics.tex396
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diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 2ded50b..bff91ef 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -107,7 +107,7 @@ the surface of the unit sphere.
Now that we have defined an operator, we can go and study its eigenfunctions,
which means that we would like to find the functions $f(\vartheta, \varphi)$
that satisfy the equation
-\begin{equation} \label{kuvel:eqn:eigen}
+\begin{equation} \label{kugel:eqn:eigen}
\surflaplacian f = -\lambda f.
\end{equation}
Perhaps it may not be obvious at first glance, but we are in fact dealing with a
@@ -178,7 +178,7 @@ write the solutions
The restriction that the separation constant $m$ needs to be an integer arises
from the fact that we require a $2\pi$-periodicity in $\varphi$ since the
coordinate systems requires that $\Phi(\varphi + 2\pi) = \Phi(\varphi)$.
-Unfortunately, solving \eqref{kugel:eqn:ode-theta} is as straightforward,
+Unfortunately, solving \eqref{kugel:eqn:ode-theta} is not as straightforward,
actually, it is quite difficult, and the process is so involved that it will
require a dedicated section of its own.
@@ -220,7 +220,7 @@ and $\lambda = n(n+1)$, we obtain what is known in the literature as the
\emph{associated Legendre equation of order $m$}:
\nocite{olver_introduction_2013}
\begin{equation} \label{kugel:eqn:associated-legendre}
- (1 - z^2)\frac{d^2 Z}{dz}
+ (1 - z^2)\frac{d^2 Z}{dz^2}
- 2z\frac{d Z}{dz}
+ \left( n(n + 1) - \frac{m^2}{1 - z^2} \right) Z(z) = 0,
\quad
@@ -236,7 +236,7 @@ This reduces the problem because it removes the double pole, which is always
tricky to deal with. In fact, the reduced problem when $m = 0$ is known as the
\emph{Legendre equation}:
\begin{equation} \label{kugel:eqn:legendre}
- (1 - z^2)\frac{d^2 Z}{dz}
+ (1 - z^2)\frac{d^2 Z}{dz^2}
- 2z\frac{d Z}{dz}
+ n(n + 1) Z(z) = 0,
\quad
@@ -250,7 +250,7 @@ case of the former that is known known as the \emph{Legendre polynomials}, since
we only need a solution between $-1$ and $1$.
\begin{lemma}[Legendre polynomials]
- \label{kugel:lem:legendre-poly}
+ \label{kugel:thm:legendre-poly}
The polynomial function
\[
P_n(z) = \sum^{\lfloor n/2 \rfloor}_{k=0}
@@ -275,7 +275,7 @@ Further, there are a few more interesting but not very relevant forms to write
$P_n(z)$ such as \emph{Rodrigues' formula} and \emph{Laplace's integral
representation} which are
\begin{equation*}
- P_n(z) = \frac{1}{2^n} \frac{d^n}{dz^n} (x^2 - 1)^n,
+ P_n(z) = \frac{1}{2^n n!} \frac{d^n}{dz^n} (z^2 - 1)^n,
\qquad \text{and} \qquad
P_n(z) = \frac{1}{\pi} \int_0^\pi \left(
z + \cos\vartheta \sqrt{z^2 - 1}
@@ -287,7 +287,7 @@ Legendre equation, we can make use of the following lemma patch the solutions
such that they also become solutions of the associated Legendre equation
\eqref{kugel:eqn:associated-legendre}.
-\begin{lemma} \label{kugel:lem:extend-legendre}
+\begin{lemma} \label{kugel:thm:extend-legendre}
If $Z_n(z)$ is a solution of the Legendre equation \eqref{kugel:eqn:legendre},
then
\begin{equation*}
@@ -300,7 +300,7 @@ such that they also become solutions of the associated Legendre equation
See section \ref{kugel:sec:proofs:legendre}.
\end{proof}
-What is happening in lemma \ref{kugel:lem:extend-legendre}, is that we are
+What is happening in lemma \ref{kugel:thm:extend-legendre}, is that we are
essentially inserting a square root function in the solution in order to be able
to reach the parts of the domain near the poles at $\pm 1$ of the associated
Legendre equation, which is not possible only using power series
@@ -312,8 +312,8 @@ obtain the \emph{associated Legendre functions}.
\label{kugel:def:ferrers-functions}
The functions
\begin{equation}
- P^m_n (z) = \frac{1}{n!2^n}(1-z^2)^{\frac{m}{2}}\frac{d^{m}}{dz^{m}} P_n(z)
- = \frac{1}{n!2^n}(1-z^2)^{\frac{m}{2}}\frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n
+ P^m_n (z) = (1-z^2)^{\frac{m}{2}}\frac{d^{m}}{dz^{m}} P_n(z)
+ = \frac{1}{2^n n!}(1-z^2)^{\frac{m}{2}}\frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n
\end{equation}
are known as Ferrers or associated Legendre functions.
\end{definition}
@@ -356,9 +356,10 @@ $Y^m_n(\vartheta, \varphi)$.
\label{kugel:def:spherical-harmonics}
The functions
\begin{equation*}
- Y_{m,n}(\vartheta, \varphi) = P^m_n(\cos \vartheta) e^{im\varphi},
+ Y^m_n (\vartheta, \varphi) = P^m_n(\cos \vartheta) e^{im\varphi},
\end{equation*}
- where $m, n \in \mathbb{Z}$ and $|m| < n$ are called spherical harmonics.
+ where $m, n \in \mathbb{Z}$ and $|m| < n$ are called (unnormalized) spherical
+ harmonics.
\end{definition}
\begin{figure}
@@ -366,45 +367,358 @@ $Y^m_n(\vartheta, \varphi)$.
\kugelplaceholderfig{\textwidth}{.8\paperheight}
\caption{
\kugeltodo{Big picture with the first few spherical harmonics.}
+ \label{kugel:fig:spherical-harmonics}
}
\end{figure}
-\subsection{Normalization}
+\kugeltodo{Describe how they look like with fig.
+\ref{kugel:fig:spherical-harmonics}}
-\kugeltodo{Discuss various normalizations.}
+\subsection{Orthogonality of $P_n$, $P^m_n$ and $Y^m_n$}
-\if 0
-As explained in the chapter \ref{}, the concept of orthogonality is very important and at the practical level it is very useful, because it allows us to develop very powerful techniques at the mathematical level.\newline
-Throughout this book we have been confronted with the Sturm-Liouville theory (see chapter \ref{}). The latter, among other things, carries with it the concept of orthogonality. Indeed, if we consider the solutions of the Sturm-Liouville equation, which can be expressed in this form
-\begin{equation}\label{kugel:eq:sturm_liouville}
- \mathcal{S}f := \frac{d}{dx}\left[p(x)\frac{df}{dx}\right]+q(x)f(x)
-\end{equation}
-possiamo dire che formano una base ortogonale.\newline
-Adesso possiamo dare un occhiata alle due equazioni che abbiamo ottenuto tramite la Separation Ansatz (Eqs.\eqref{kugel:eq:associated_leg_eq}\eqref{kugel:eq:ODE_1}), le quali possono essere riscritte come:
-\begin{align*}
- \frac{d}{dx} \left[ (1-x^2) \cdot \frac{dP_{m,n}}{dx} \right] &+ \left(n(n+1)-\frac{m}{1-x^2} \right) \cdot P_{m,n}(x) = 0, \\
- \frac{d}{d\varphi} \left[ 1 \cdot \frac{ d\Phi }{d\varphi} \right] &+ 1 \cdot \Phi(\varphi) = 0.
-\end{align*}
-Si può concludere in modo diretto che sono due casi dell'equazione di Sturm-Liouville. Questo significa che le loro soluzioni sono ortogonali sotto l'inner product con weight function $w(x)=1$, dunque:
-\begin{align}
-\int_{0}^{2\pi} \Phi_m(\varphi)\Phi_m'(\varphi) d\varphi &= \delta_{m'm}, \nonumber \\
-\int_{-1}^1 P_{m,m'}(x)P_{n,n'}(x) dx &= \delta_{m'm}\delta_{n'n}. \label{kugel:eq:orthogonality_associated_func}
-\end{align}
-Inoltre, possiamo provare l'ortogonalità di $\Theta(\vartheta)$ utilizzando \eqref{kugel:eq:orthogonality_associated_func}:
-\begin{align}
- x
-\end{align}
-Ora, visto che la soluzione dell'eigenfunction problem è formata dalla moltiplicazione di $\Phi_m(\varphi)$ e $P_{m,n}(x)$
-\fi
+We shall now discuss an important property of the spherical harmonics: they form
+an orthogonal system. And since the spherical harmonics contain the Ferrers or
+associated Legendre functions, we need to discuss their orthogonality first.
+But the Ferrers functions themselves depend on the Legendre polynomials, so that
+will be our starting point.
-\subsection{Properties}
+\begin{lemma} For the Legendre polynomials $P_n(z)$ and $P_k(z)$ it holds that
+ \label{kugel:thm:legendre-poly-ortho}
+ \begin{equation*}
+ \int_{-1}^1 P_n(z) P_k(z) \, dz
+ = \frac{2}{2n + 1} \delta_{nk}
+ = \begin{cases}
+ \frac{2}{2n + 1} & \text{if } n = k, \\
+ 0 & \text{otherwise}.
+ \end{cases}
+ \end{equation*}
+\end{lemma}
+\begin{proof}
+ To start, consider the fact that the Legendre equation
+ \eqref{kugel:eqn:legendre}, of which two distinct Legendre polynomials
+ $P_n(z)$ and $P_k(z)$ are a solution ($n \neq k$), can be rewritten in the
+ following form:
+ \begin{equation}
+ \frac{d}{dz} \left[
+ \left( 1 - z^2 \right) \frac{dZ}{dz}
+ \right] + n(n+1) Z(z) = 0.
+ \end{equation}
+ So we rewrite the Legendre equations for $P_n(z)$ and $P_k(z)$:
+ \begin{align*}
+ \frac{d}{dz} \left[
+ \left( 1 - z^2 \right) \frac{dP_n}{dz}
+ \right] + n(n+1) P_n(z) &= 0,
+ &
+ \frac{d}{dz} \left[
+ \left( 1 - z^2 \right) \frac{dP_k}{dz}
+ \right] + k(k+1) P_k(z) &= 0,
+ \end{align*}
+ then we multiply the former by $P_k(z)$ and the latter by $P_n(z)$ and
+ subtract the two to get
+ \begin{equation*}
+ \frac{d}{dz} \left[
+ \left( 1 - z^2 \right) \frac{dP_n}{dz}
+ \right] P_k(z) + n(n+1) P_n(z) P_k(z)
+ -
+ \frac{d}{dz} \left[
+ \left( 1 - z^2 \right) \frac{dP_k}{dz}
+ \right] P_n(z) - k(k+1) P_k(z) P_n(z) = 0.
+ \end{equation*}
+ By grouping terms, making order and integrating with respect to $z$ from $-1$
+ to 1 we obtain
+ \begin{gather}
+ \int_{-1}^1 \left\{
+ \frac{d}{dz} \left[
+ \left( 1 - z^2 \right) \frac{dP_n}{dz}
+ \right] P_k(z)
+ -
+ \frac{d}{dz} \left[
+ \left( 1 - z^2 \right) \frac{dP_k}{dz}
+ \right] P_n(z) - k(k+1) P_k(z) P_n(z)
+ \right\} \,dz \nonumber \\
+ + \left[ n(n+1) - k(k+1) \right] \int_{-1}^1 P_k(z) P_n(z) \, dz = 0.
+ \label{kugel:thm:legendre-poly-ortho:proof:1}
+ \end{gather}
+ Since by the product rule
+ \begin{equation*}
+ \frac{d}{dz} \left[ (1 - z^2) \frac{dP_k}{dz} P_n(z) \right]
+ =
+ \frac{d}{dz} \left[ (1 - z^2) \frac{dP_n}{dz} \right] P_k(z)
+ + (1 - z^2) \frac{dP_n}{dz} \frac{dP_k}{dz},
+ \end{equation*}
+ we can simplify the first term in
+ \eqref{kugel:thm:legendre-poly-ortho:proof:1} to get
+ \begin{gather*}
+ \int_{-1}^1 \left\{
+ \frac{d}{dz} \left[ (1 - z^2) \frac{dP_k}{dz} P_n(z) \right]
+ - \cancel{(1 - z^2) \frac{dP_n}{dz} \frac{dP_k}{dz}}
+ - \frac{d}{dz} \left[ (1 - z^2) \frac{dP_n}{dz} P_k(z) \right]
+ + \cancel{(1 - z^2) \frac{dP_k}{dz} \frac{dP_n}{dz}}
+ \right\} \, dz \\
+ = \int_{-1}^1 \frac{d}{dz} \left\{ (1 - z^2) \left[
+ \frac{dP_k}{dz} P_n(z) - \frac{dP_n}{dz} P_k(z)
+ \right] \right\} \, dz
+ = (1 - z^2) \left[
+ \frac{dP_k}{dz} P_n(z) - \frac{dP_n}{dz} P_k(z)
+ \right] \Bigg|_{-1}^1,
+ \end{gather*}
+ which always equals 0 because the product contains $1 - z^2$ and the bounds
+ are at $\pm 1$. Thus, of \eqref{kugel:thm:legendre-poly-ortho:proof:1} only
+ the second term remains and the equation becomes
+ \begin{equation*}
+ \left[ n(n+1) - k(k+1) \right] \int_{-1}^1 P_k(z) P_n(z) \, dz = 0.
+ \end{equation*}
+ By dividing by the constant in front of the integral we have our first result.
+ Now we need to show that when $n = k$ the integral equals $2 / (2n + 1)$.
+ % \begin{equation*}
+ % \end{equation*}
+ \kugeltodo{Finish proof. Can we do it without the generating function of
+ $P_n$?}
+\end{proof}
+
+In a similarly algebraically tedious fashion, we can also continue to check for
+orthogonality for the Ferrers functions $P^m_n(z)$, since they are related to
+$P_n(z)$ by a $m$-th derivative, and obtain the following result.
+
+\begin{lemma} For the associated Legendre functions
+ \label{kugel:thm:associated-legendre-ortho}
+ \begin{equation*}
+ \int_{-1}^1 P^m_n(z) P^{m}_{n'}(z) \, dz
+ = \frac{2(m + n)!}{(2n + 1)(n - m)!} \delta_{nn'}
+ = \begin{cases}
+ \frac{2(m + n)!}{(2n + 1)(n - m)!}
+ & \text{if } n = n', \\
+ 0 & \text{otherwise}.
+ \end{cases}
+ \end{equation*}
+\end{lemma}
+\begin{proof}
+ To show that the expression equals zero when $n \neq n'$ we can perform
+ exactly the same steps as in the proof of lemma
+ \ref{kugel:thm:legendre-poly-ortho}, so we will not repeat them here and prove
+ instead only the case when $n = n'$.
+ \kugeltodo{Finish proof, or not? I have to look and decide if it is
+ interesting enough.}
+\end{proof}
+
+By having the orthogonality relations of the Legendre functions we can finally
+show that spherical harmonics are also orthogonal under the following inner
+product:
+
+\begin{definition}[Inner product in $S^2$]
+ \label{kugel:def:inner-product-s2}
+ For 2 complex valued functions $f(\vartheta, \varphi)$ and $g(\vartheta,
+ \varphi)$ on the surface of the sphere the inner product is defined to be
+ \begin{equation*}
+ \langle f, g \rangle
+ = \int_{0}^\pi \int_0^{2\pi}
+ f(\vartheta, \varphi) \overline{g(\vartheta, \varphi)}
+ \sin \vartheta \, d\varphi \, d\vartheta.
+ \end{equation*}
+\end{definition}
+
+
+\begin{theorem} For the (unnormalized) spherical harmonics
+ \label{kugel:thm:spherical-harmonics-ortho}
+ \begin{align*}
+ \langle Y^m_n, Y^{m'}_{n'} \rangle
+ &= \int_{0}^\pi \int_0^{2\pi}
+ Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)}
+ \sin \vartheta \, d\varphi \, d\vartheta
+ \\
+ &= \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} \delta_{nn'} \delta_{mm'}
+ = \begin{cases}
+ \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!}
+ & \text{if } n = n' \text{ and } m = m', \\
+ 0 & \text{otherwise}.
+ \end{cases}
+ \end{align*}
+\end{theorem}
+\begin{proof}
+ We will begin by doing a bit of algebraic maipulaiton:
+ \begin{align*}
+ \int_{0}^\pi \int_0^{2\pi}
+ Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)}
+ \sin \vartheta \, d\varphi \, d\vartheta
+ &= \int_{0}^\pi \int_0^{2\pi}
+ e^{im\varphi} P^m_n(\cos \vartheta)
+ e^{-im'\varphi} P^{m'}_{n'}(\cos \vartheta)
+ \, d\varphi \sin \vartheta \, d\vartheta
+ \\
+ &= \int_{0}^\pi
+ P^m_n(\cos \vartheta) P^{m'}_{n'}(\cos \vartheta)
+ \int_0^{2\pi} e^{i(m - m')\varphi}
+ \, d\varphi \sin \vartheta \, d\vartheta
+ .
+ \end{align*}
+ First, notice that the associated Legendre polynomials are assumed to be real,
+ and are thus unaffected by the complex conjugation. Then, we can see that when
+ $m = m'$ the inner integral simplifies to $\int_0^{2\pi} 1 \, d\varphi$ which
+ equals $2\pi$, so in this case the expression becomes
+ \begin{equation*}
+ 2\pi \int_{0}^\pi
+ P^m_n(\cos \vartheta) P^{m'}_{n'}(\cos \vartheta)
+ \sin \vartheta \, d\vartheta
+ = -2\pi \int_{1}^{-1} P^m_n(z) P^{m'}_{n'}(z) \, dz
+ = \frac{4\pi(m + n)!}{(2n + 1)(n - m)!} \delta_{nn'},
+ \end{equation*}
+ where in the second step we performed the substitution $z = \cos\vartheta$;
+ $d\vartheta = \frac{d\vartheta}{dz} dz= - dz / \sin \vartheta$, and then we
+ used lemma \ref{kugel:thm:associated-legendre-ortho}. We are allowed to use
+ the lemma because $m = m'$.
+
+ Now we just need look at the case when $m \neq m'$. Fortunately this is
+ easier: the inner integral is $\int_0^{2\pi} e^{i(m - m')\varphi} d\varphi$,
+ or in other words we are integrating a complex exponetial over the entire
+ period, which always results in zero. Thus, we do not need to do anything and
+ the proof is complete.
+\end{proof}
+
+These proofs for the various orthogonality relations were quite long and
+algebraically tedious, mainly because they are ``low level'', by which we mean
+that they (arguably) do not rely on very abstract theory. However, if we allow
+ourselves to use the more abstract Sturm Liouville theory discussed in chapters
+\ref{buch:integrale:subsection:sturm-liouville-problem} and \kugeltodo{reference
+to chapter 17 of haddouche and Löffler} the proofs can become ridiculously
+short. Let's do for example lemma \ref{kugel:thm:associated-legendre-ortho}.
+
+\begin{proof}[
+ Shorter proof of lemma \ref{kugel:thm:associated-legendre-ortho}
+ ]
+ The associated Legendre polynomials, of which we would like to prove an
+ orthogonality relation, are the solution to the associated Legendre equation,
+ which we can write as $LZ(z) = 0$, where
+ \begin{equation*}
+ L = \frac{d}{dz} (1 - z^2) \frac{d}{dz}
+ + n(n+1) - \frac{m^2}{1 - z^2}.
+ \end{equation*}
+ Notice that $L$ is in fact a Sturm-Liouville operator of the form
+ \begin{equation*}
+ L = \frac{1}{w(z)} \left[
+ \frac{d}{dz} p(z) \frac{d}{dz} - \lambda + q(z)
+ \right],
+ \end{equation*}
+ if we let $w(z) = 1$, $p(z) = (1 - z^2 )$, $q(z) = -m^2 / (1 - z^2)$, and
+ $\lambda = -n(n+1)$. By the theory of Sturm-Liouville operators, we know that
+ the each solution of the problem $LZ(z) = 0$, namely $P^m_n(z)$, is orthogonal
+ to every other solution that has a different $\lambda$. In our case $\lambda$
+ varies with $n$, so $P^m_n(z)$ with different $n$'s are orthogonal to each
+ other.
+\end{proof}
+
+But that was still rather informative and had a bit of explanation, which is
+terrible. Real snobs, such as Wikipedia contributors, some authors and
+regrettably sometimes even ourselves, would write instead:
+
+\begin{proof}[
+ Infuriatingly short proof of lemma \ref{kugel:thm:associated-legendre-ortho}
+ ]
+ The associated Legendre polynomials are solutions of the associated Legendre
+ equation which is a Sturm-Liouville problem and are thus orthogonal to each
+ other. The factor in front Kronecker delta is left as an exercise to the
+ reader.
+\end{proof}
+
+Lemma \ref{kugel:thm:legendre-poly-ortho} has a very similar
+proof, while the theorem \ref{kugel:thm:spherical-harmonics-ortho} for the
+spherical harmonics is proved by the following argument. The spherical harmonics
+are the solutions to the eigenvalue problem $\surflaplacian f = -\lambda f$,
+which as discussed in the previous section is solved using separation. So to
+prove their orthogonality using the Sturm-Liouville theory we argue that
+\begin{equation*}
+ \surflaplacian = L_\vartheta L_\varphi \iff
+ \surflaplacian f(\vartheta, \varphi)
+ = L_\vartheta \Theta(\vartheta) L_\varphi \Phi(\varphi),
+\end{equation*}
+then we show that both $L_\vartheta$ and $L_\varphi$ are both Sturm-Liouville
+operators (we just did the former in the shorter proof above). Since both are
+Sturm-Liouville operators their combination, the surface spherical Laplacian, is
+also a Sturm-Liouville operator, which then implies orthogonality.
+
+\subsection{Normalization and the Phase Factor}
+
+At this point we have shown that the spherical harmonics form an orthogonal
+system, but in many applications we usually also want a normalization of some
+kind. For example the most obvious desirable property could be for the spherical
+harmonics to be ortho\emph{normal}, by which we mean that $\langle Y^m_n,
+Y^{m'}_{n'} \rangle = \delta_{nn'}$. To obtain orthonormality, we simply add an
+ugly normalization factor in front of the previous definition
+\ref{kugel:def:spherical-harmonics} as follows.
+
+\begin{definition}[Orthonormal spherical harmonics]
+ \label{kugel:def:spherical-harmonics-orthonormal}
+ The functions
+ \begin{equation*}
+ Y^m_n(\vartheta, \varphi)
+ = \sqrt{\frac{2n + 1}{4\pi} \frac{(n-m)!}{(m+n)!}}
+ P^m_n(\cos \vartheta) e^{im\varphi}
+ \end{equation*}
+ where $m, n \in \mathbb{Z}$ and $|m| < n$ are the orthonormal spherical
+ harmonics.
+\end{definition}
+
+Orthornomality is very useful, but it is not the only common normalization that
+is found in the literature. In physics, geomagnetism to be more specific, it is
+common to use the so called Schmidt semi-normalization (or sometimes also called
+quasi-normalization).
+
+\begin{definition}[Schmidt semi-normalized spherical harmonics]
+ \label{kugel:def:spherical-harmonics-schmidt}
+ The Schmidt semi-normalized spherical harmonics are
+ \begin{equation*}
+ Y^m_n(\vartheta, \varphi)
+ = \sqrt{2 \frac{(n - m)!}{(n + m)!}}
+ P^m_n(\cos \vartheta) e^{im\varphi}
+ \end{equation*}
+ where $m, n \in \mathbb{Z}$ and $|m| < n$.
+\end{definition}
+
+Additionally, there is another quirk in the literature that should be mentioned.
+In some other branches of physics such as seismology and quantum mechanics there
+is a so called Condon-Shortley phase factor $(-1)^m$ in front of the square root
+in the definition of the normalized spherical harmonics. It is yet another
+normalization that is added for physical reasons that are not very relevant to
+our discussion, but we mention this potential source of confusion since many
+numerical packages (such as \texttt{SHTOOLS} \kugeltodo{Reference}) offer an
+option to add or remove it from the computation.
+
+Though, for our purposes we will mostly only need the orthonormal spherical
+harmonics, so from now on, unless specified otherwise when we say spherical
+harmonics or write $Y^m_n$, we mean the orthonormal spherical harmonics of
+definition \ref{kugel:def:spherical-harmonics-orthonormal}.
\subsection{Recurrence Relations}
-\section{Series Expansions in $C(S^2)$}
+\section{Series Expansions in $L^2(S^2)$}
-\subsection{Orthogonality of $P_n$, $P^m_n$ and $Y^m_n$}
+We have now reached a point were we have all of the tools that are necessary to
+build something truly amazing: a general series expansion formula for functions
+on the surface of the sphere. Using the jargon: we will now see that the
+spherical harmonics together with the inner product of definition
+\ref{kugel:def:inner-product-s2}
+\begin{equation*}
+ \langle f, g \rangle
+ = \int_{0}^\pi \int_0^{2\pi}
+ f(\vartheta, \varphi) \overline{g(\vartheta, \varphi)}
+ \sin \vartheta \, d\varphi \, d\vartheta
+\end{equation*}
+form a Hilbert space over the space of complex valued $L^2$ functions $S^2 \to
+\mathbb{C}$. We will see later that this fact is very consequential and is
+extremely useful for many types of applications. If the jargon was too much, no
+need to worry, we will now go back to normal words and explain it again in more
+detail.
+
+\subsection{Spherical Harmonics Series}
-\subsection{Series Expansion}
+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]
+ \begin{equation*}
+ \hat{f}(\vartheta, \varphi)
+ = \sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}}
+ c_{m,n} Y^m_n(\vartheta, \varphi)
+ \end{equation*}
+\end{definition}
\subsection{Fourier on $S^2$}