From d2ae59bb9d2affc07bcb541d37a8f88fd009c167 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sat, 20 Aug 2022 19:49:13 +0200
Subject: kugel: mention Condon-Shortley phase factor

---
 buch/papers/kugel/spherical-harmonics.tex | 13 +++++++++++--
 1 file changed, 11 insertions(+), 2 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 72f7402..5d394a9 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -639,8 +639,17 @@ quasi-normalization).
   where $m, n \in \mathbb{Z}$ and $|m| < n$.
 \end{definition}
 
-However, for our purposes we will mostly only need the orthonormal spherical
-harmonics.  So from now on, unless specified otherwise, when we say spherical
+Additionally, there is another quirk in the literature that should be mentioned.
+In some other branches of physics such as seismology 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 reasons that are not very relevant to our
+discussion, but we are mentioning its existence 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}.
 
-- 
cgit v1.2.1


From f05ad8165a516c7932a8137a51b247484c38403b Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sat, 20 Aug 2022 23:25:12 +0200
Subject: kugel: Orthogonality using Sturm-Liouville

---
 buch/papers/kugel/spherical-harmonics.tex | 92 +++++++++++++++++++++++++++----
 1 file changed, 82 insertions(+), 10 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 5d394a9..5a17b99 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -506,6 +506,7 @@ 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*}
@@ -573,8 +574,51 @@ product:
   the proof is complete.
 \end{proof}
 
-\kugeltodo{Briefly mention that we could have skipped the tedious proofs by
-showing that the (associated) Legendre equation is a Sturm Liouville problem.}
+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 sometimes
+regrettably even ourselves, would write instead:
+
+\begin{proof}[
+    Pretentiously 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}
+
 
 \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 
@@ -640,13 +684,13 @@ quasi-normalization).
 \end{definition}
 
 Additionally, there is another quirk in the literature that should be mentioned.
-In some other branches of physics such as seismology 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 reasons that are not very relevant to our
-discussion, but we are mentioning its existence since many numerical packages
-(such as \texttt{SHTOOLS} \kugeltodo{Reference}) offer an option to add or
-remove it from the computation.
+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 mention its existence 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
@@ -655,8 +699,36 @@ definition \ref{kugel:def:spherical-harmonics-orthonormal}.
 
 \subsection{Recurrence Relations}
 
-\section{Series Expansions in $C(S^2)$}
+\section{Series Expansions in $L^2(S^2)$}
+
+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}
 
+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 $S^2$}
-- 
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From 63dee97e79f65a967f7d6b34bb8141ccaa226e20 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sat, 20 Aug 2022 23:40:29 +0200
Subject: kugel: Minor corrections

---
 buch/papers/kugel/spherical-harmonics.tex | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 5a17b99..54c8fa9 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -607,11 +607,11 @@ short. Let's do for example lemma \ref{kugel:thm:associated-legendre-ortho}.
 \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 sometimes
-regrettably even ourselves, would write instead:
+terrible. Real snobs, such as Wikipedia contributors, some authors and
+regrettably sometimes even ourselves, would write instead:
 
 \begin{proof}[
-    Pretentiously short proof of lemma \ref{kugel:thm:associated-legendre-ortho}
+    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
@@ -688,9 +688,9 @@ 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 mention its existence 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.
+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
-- 
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From 288eb54f5089c48177434757b083309e05e30bf2 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sun, 21 Aug 2022 11:48:48 +0200
Subject: kugel: More on Sturm-Liouville

---
 buch/papers/kugel/spherical-harmonics.tex | 42 ++++++++++++-------------------
 1 file changed, 16 insertions(+), 26 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 54c8fa9..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
@@ -619,31 +619,21 @@ regrettably sometimes even ourselves, would write instead:
   reader.
 \end{proof}
 
-
-\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
-
+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}
 
-- 
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