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diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex index bff91ef..9349b61 100644 --- a/buch/papers/kugel/spherical-harmonics.tex +++ b/buch/papers/kugel/spherical-harmonics.tex @@ -111,7 +111,10 @@ that satisfy the equation \surflaplacian f = -\lambda f. \end{equation} Perhaps it may not be obvious at first glance, but we are in fact dealing with a -partial differential equation (PDE) \kugeltodo{Boundary conditions?}. If we +partial differential equation (PDE)\footnote{ + Considering the fact that we are dealing with a PDE, + you may be wondering what are the boundary conditions. Well, since this eigenvalue problem is been developed on + the spherical surface (boundary of a sphere), the boundary in this case are empty, i.e no boundary condition has to be considered.}. unpack the notation of the operator $\nabla^2_{\partial S}$ according to definition \ref{kugel:def:surface-laplacian}, we get: @@ -283,7 +286,7 @@ representation} which are \end{equation*} respectively, both of which we will not prove (see chapter 3 of \cite{bell_special_2004} for a proof). Now that we have a solution for the -Legendre equation, we can make use of the following lemma patch the solutions +Legendre equation, we can make use of the following lemma to patch the solutions such that they also become solutions of the associated Legendre equation \eqref{kugel:eqn:associated-legendre}. @@ -313,24 +316,19 @@ obtain the \emph{associated Legendre functions}. The functions \begin{equation} 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 + = \frac{1}{2^n n!}(1-z^2)^{\frac{m}{2}}\frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n, \quad |m|<n \end{equation} are known as Ferrers or associated Legendre functions. \end{definition} +The constraint $|m|<n$, can be justified by considering eq.\eqref{kugel:eq:associated_leg_func}, where we differentiate $m+n$ times. We all know that a differentiation, to be well defined, must have an order that is greater than zero \kugeltodo{is that always true?}. Furthermore, it can be seen that this derivative is applied on a polynomial of degree $2n$. As is known from Calculus 1, if you derive a polynomial of degree $2n$ more than $2n$ times, you get zero, that would be a trivial solution. This is the power of zero: It is almost always a (boring) solution. -\kugeltodo{Discuss $|m| \leq n$.} - -\if 0 -The constraint $|m|<n$, can be justified by considering Eq.\eqref{kugel:eq:associated_leg_func}, in which the derivative of degree $m+n$ is present. A derivative to be well defined must have an order that is greater than zero. Furthermore, it can be seen that this derivative is applied on a polynomial of degree $2n$. As is known from Calculus 1, if you derive a polynomial of degree $2n$ more than $2n$ times, you get zero, which is a trivial solution in which we are not interested.\newline We can thus summarize these two conditions by writing: \begin{equation*} \begin{rcases} m+n \leq 2n &\implies m \leq n \\ m+n \geq 0 &\implies m \geq -n - \end{rcases} |m| \leq n. + \end{rcases} \; |m| \leq n. \end{equation*} -The set of functions in Eq.\eqref{kugel:eq:sph_harm_0} is named \emph{Spherical Harmonics}, which are the eigenfunctions of the Laplace operator on the \emph{spherical surface domain}, which is exactly what we were looking for at the beginning of this section. -\fi \subsection{Spherical Harmonics} @@ -339,13 +337,13 @@ section \ref{kugel:sec:construction:eigenvalue}. We had left off in the middle of the separation, were we had used the Ansatz $f(\vartheta, \varphi) = \Theta(\vartheta) \Phi(\varphi)$ to find that $\Phi(\varphi) = e^{im\varphi}$, and we were solving for $\Theta(\vartheta)$. As you may recall, previously we -performed the substitution $z = \cos \vartheta$. Now we can finally to bring back the +performed the substitution $z = \cos \vartheta$. Now we can finally bring back the solution to the associated Legendre equation $P^m_n(z)$ into the $\vartheta$ domain and combine it with $\Phi(\varphi)$ to get the full result: \begin{equation*} f(\vartheta, \varphi) = \Theta(\vartheta)\Phi(\varphi) - = P^m_n (\cos \vartheta) e^{im\varphi}. + = P^m_n (\cos \vartheta) e^{im\varphi}, \quad |m|<n. \end{equation*} This family of functions, which recall are the solutions of the eigenvalue problem of the surface spherical Laplacian, are the long anticipated @@ -356,9 +354,9 @@ $Y^m_n(\vartheta, \varphi)$. \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}, \quad |m|<n \end{equation*} - where $m, n \in \mathbb{Z}$ and $|m| < n$ are called (unnormalized) spherical + where $m, n \in \mathbb{Z}$ are called (unnormalized) spherical harmonics. \end{definition} @@ -507,7 +505,7 @@ 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, + For two 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 @@ -520,36 +518,35 @@ product: \begin{theorem} For the (unnormalized) spherical harmonics \label{kugel:thm:spherical-harmonics-ortho} - \begin{align*} + \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 - \\ + \label{kugel:eq:spherical-harmonics-inner-prod} \\ &= \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', \\ + & \text{if } n = n' \text{ and } m = m', \nonumber \\ 0 & \text{otherwise}. \end{cases} - \end{align*} + \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)} + 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 + \, d\varphi \sin \vartheta \, d\vartheta \\ &= \int_{0}^\pi - P^m_n(\cos \vartheta) P^{m'}_{n'}(\cos \vartheta) + P^m_n(\cos \vartheta) P^{m'}_{n'}(\cos \vartheta) \sin \vartheta \, d\vartheta \int_0^{2\pi} e^{i(m - m')\varphi} - \, d\varphi \sin \vartheta \, d\vartheta - . + \, d\varphi. \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 @@ -564,12 +561,15 @@ product: \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 $m = m'$. - + used lemma \ref{kugel:thm:associated-legendre-ortho}. + We are allowed to use + the lemma because $m = m'$. After the just mentioned substitution we can write eq.\eqref{kugel:eq:spherical-harmonics-inner-prod} in this form + \begin{equation*} + \langle Y^m_n, Y^{m'}_{n'} \rangle_{\partial S} = \langle P^m_n, P^{m'}_{n'} \rangle_z \; \langle e^{im\varphi}, e^{-im'\varphi} \rangle_\varphi. + \end{equation*} 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 + or in other words we are integrating a complex exponential over the entire period, which always results in zero. Thus, we do not need to do anything and the proof is complete. \end{proof} @@ -619,11 +619,9 @@ regrettably sometimes even ourselves, would write instead: 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 +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 the separation Ansatz. So to prove their orthogonality using the Sturm-Liouville theory we argue that \begin{equation*} \surflaplacian = L_\vartheta L_\varphi \iff @@ -687,26 +685,196 @@ 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} +\subsection{Recurrence Relations}\kugeltodo{replace x with z} +The idea of this subsection is to introduce first some recursive relations regarding the Associated Legendre Functions, defined in eq.\eqref{kugel:def:ferrers-functions}. Subsequently we will extend them, in order to derive recurrence formulas for the case of Spherical Harmonic functions as well. +\subsubsection{Associated Legendre Functions} +To start this journey, we can first write the following equations, which relate the Associated Legendre functions of different indeces $m$ and $n$ recursively: +\begin{subequations} + \begin{align} + P^m_n(z) &= \dfrac{1}{(2n+1)x} \left[ (m+n) P^m_{n-1}(z) + (n-m+1) P^m_{n+1}(z) \right] \label{kugel:eq:rec-leg-1} \\ + P^m_n(z) &= \dfrac{\sqrt{1-z^2}}{2mz} \left[ P^{m+1}_n(z) + [n(n+1)-m(m-1)] P^{m-1}_n(z) \right] \label{kugel:eq:rec-leg-2} \\ + P^m_n(z) &= \dfrac{1}{(2n+1)\sqrt{1-z^2}} \left[ P^{m+1}_{n+1}(z) - P^{m+1}_{n-1}(z) \right] \label{kugel:eq:rec-leg-3} \\ + P^m_n(z) &= \dfrac{1}{(2n+1)\sqrt{1-z^2}} \left[ (n+m)(n+m-1)P^{m-1}_{n-1}(z) - (n-m+1)(n-m+2)P^{m-1}_{n+1}(z) \right] \label{kugel:eq:rec-leg-4} + \end{align} +\end{subequations} +Much of the effort will be proving this bunch of equalities. Then, in the second 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 softwares, 3-D 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{usecase_recursion_paper}. +\begin{enumerate}[(i)] + \item + \begin{proof} + This is the relation that links the associated Legendre functions with the same $m$ index but different $n$. Using \ref{} \kugeltodo{search the general equation of recursion for orthogonal polynomials (is somewhere in the book)}, we have + \begin{equation*} + (n+1)P_{n+1}(z)-(2n+1)xP_n(z)+nP_{n-1}(z)=0, + \end{equation*} + that can be differentiated $m$ times, obtaining + \begin{equation}\label{kugel:eq:rec_1} + (n+1)\frac{d^mP_{n+1}}{dz^m}-(2n+1) \left[z \frac{d^m P_n}{dz^m}+ m\frac{d^{m-1}P_{n-1}}{dz^{m-1}} \right] + n\frac{d^m P_{n-1}}{dz^m}=0. + \end{equation} + To continue this derivation, we need the following relation: + \begin{equation}\label{kugel:eq:rec_2} + \frac{dP_{n+1}}{dz} - \frac{dP_{n-1}}{dz} = (2n+1)P_n. + \end{equation} + The latter will not be derived, because it suffices to use the definition of the Legendre Polynomials $P_n(x)$ to check it. + + We can now differentiate the just presented eq.\eqref{kugel:eq:rec_2} $m-1$ times, that will become + \begin{equation}\label{kugel:eq:rec_3} + \frac{d^mP_{n+1}}{dx^m} - \frac{d^mP_{n-1}}{dx^m} = (2n+1)\frac{d^{m-1}P_n}{dx^{m-1}}. + \end{equation} + Then, using eq.\eqref{kugel:eq:rec_3} in eq.\eqref{kugel:eq:rec_1}, we will have + \begin{equation}\label{kugel:eq:rec_4} + (n+1)\frac{d^mP_{n+1}}{dx^m}- (2n+1)\frac{d^mP_{n+1}}{dx^m} -m\left[\frac{d^m P_{n+1}}{dx^m}+ \frac{d^{m}P_{n-1}}{dx^m}\right] + n\frac{d^m P_{n-1}}{dx^m}=0. + \end{equation} + Finally, multiplying both sides by $(1-x^2)^{\frac{m}{2}}$ and simplifying the expression, we can rewrite eq.\eqref{kugel:eq:rec_4} in terms of $P^m_n(x)$, namely + \begin{equation*} + (n+1-m)P^m_{n+1}(x)-(2n+1)xP^m_n(x)+(m+n)P^m_{n-1}(x)=0, + \end{equation*} + that rearranged, will be + \begin{equation*} + (2n+1) x P^m_n(x)= (m+n) P^m_{n-1}(x) + (n-m+1) P^m_{n+1}(x). + \end{equation*} + \end{proof} + + \item + \begin{proof} + This relation, unlike the previous one, link three expression with the same $n$ index but different $m$. + + In the proof of Lemma \ref{kugel:lemma:sol_associated_leg_eq}, at some point we ran into this expression. + \begin{equation*} + (1-x^2)\frac{d^{m+2}P_n}{dx^{m+2}} - 2(m+1)x \frac{d^{m+1}P_n}{dx^{m+1}} + [n(n+1)-m(m+1)]\frac{d^mP_n}{dx^m} = 0, + \end{equation*} + that, if multiplied by $(1-x^2)^{\frac{m}{2}}$, will be + \begin{equation*} + (1-x^2)^{\frac{m}{2}+1}\frac{d^{m+2}P_n}{dx^{m+2}} - 2(m+1)x (1-x^2)^{\frac{m}{2}}\frac{d^{m+1}P_n}{dx^{m+1}} + [n(n+1)-m(m+1)](1-x^2)^{\frac{m}{2}}\frac{d^mP_n}{dx^m} = 0. + \end{equation*} + Therefore, as before, expressing it in terms of $P^m_n(x)$: + \begin{equation*} + P^{m+2}_n(x) - \frac{2(m+1)x}{\sqrt{1-x^2}}P^{m+1}_n(x) + [n(n+1)-m(m+1)]P^m_n(x)=0. + \end{equation*} + Further, we can adjust the indeces and terms, obtaining + \begin{equation*} + \frac{2mx}{\sqrt{(1-x^2)}} P^m_n(x) = P^{m+1}_n(x) + [n(n+1)-m(m-1)] P^{m-1}_n(x). + \end{equation*} + + \end{proof} + + \item + \begin{proof} + To derive this expression, we can multiply eq.\eqref{kugel:eq:rec_3} by $(1-x^2)^{\frac{m}{2}}$ and, as always, we could express it in terms of $P^m_n(x)$: + \begin{equation*} + P^m_{n+1}(x) - P^m_{n-1}(x) = (2n+1)\sqrt{1-x^2}P^{m-1}_n(x). + \end{equation*} + Afer that we can divide by $2n+1$ resulting in + \begin{equation}\label{kugel:eq:helper} + \frac{1}{2n+1}[P^m_{n+1}(x) - P^m_{n-1}(x)] = \sqrt{1-x^2}P^{m-1}_n(x). + \end{equation} + To conclude, we arrange the indeces differently: + \begin{equation*} + \sqrt{1-x^2}P^{m}_n(x)=\frac{1}{2n+1}[P^{m+1}_{n+1}(x) - P^{m+1}_{n-1}(x)]. + \end{equation*} + \end{proof} + + \item + \begin{proof} + For this proof we can rely on eq.\eqref{kugel:eq:rec-leg-1}, and therefore rewrite eq.\eqref{kugel:eq:rec-leg-2} as + \begin{equation*} + \frac{2m}{(2n+1)\sqrt{1-x^2}} \left[ (m+n)P^m_{n-1}(x) + (n-m+1)P^m_{n+1}(x) \right] = P^{m+1}_n(x) + [ n(n+1)-m(m-1) ]P^{m-1}_n(x). + \end{equation*} + Rewriting then $P^{m-1}_n(x)$ using eq.\eqref{kugel:eq:helper}, we will have + \begin{align*} + \frac{2m}{(2n+1)\sqrt{1-x^2}} &\left[ (m+n)P^m_{n-1}(x) + (n-m+1)P^m_{n+1}(x) \right] = P^{m+1}_n(x) \\ + &+ \frac{n(n+1)-m(m-1)}{(2n+1)\sqrt{1-x^2}} \left[ P^m_{n+1}(x)-P^m_{n-1}(x) \right]. + \end{align*} + The last equation, after some algebric rearrangements, it is easy to show that it is equivalent to + \begin{equation*} + \sqrt{1-x^2} P^m_n(x) = \dfrac{1}{2n+1} \left[ (n+m)(n+m-1)P^{m-1}_{n-1}(x) - (n-m+1)(n-m+2)P^{m-1}_{n+1}(x) \right] + \end{equation*} + \end{proof} + +\end{enumerate} + +\subsubsection{Spherical Harmonics} +The goal of this subsection's part is to apply the recurrence relations of the $P^m_n(z)$ functions to the Spherical Harmonics. +With some little adjustments we will be able to have recursion equations for them too. As previously written the most of the work is already done. Now it is only a matter of minor mathematical operations/rearrangements. + +We can start by listing all of them: +\begin{subequations} + \begin{align} + Y^m_n(\vartheta, \varphi) &= \dfrac{1}{(2n+1)\cos \vartheta} \left[ (m+n)Y^m_{n-1}(\vartheta, \varphi) + (m-n+1)Y^m_{n+1}(\vartheta, \varphi) \right] \label{kugel:eq:rec-sph_harm-1} \\ + Y^m_n(\vartheta, \varphi) &= \dfrac{\tan \vartheta}{2m}\left[ Y^{m+1}_n(\vartheta, \varphi)e^{-i\varphi} + [n(n+1)-m(m-1)]Y^{m-1}_n(\vartheta, \varphi)e^{i\varphi} \right] \label{kugel:eq:rec-sph_harm-2} \\ + Y^m_n(\vartheta, \varphi) &= \dfrac{e^{-i\varphi}}{ (2n+1)\sin \vartheta } \left[ Y^{m+1}_{n+1}(\vartheta, \varphi) - Y^{m+1}_{n-1}(\vartheta, \varphi) \right] \label{kugel:eq:rec-sph_harm-3} \\ + Y^m_n(\vartheta, \varphi) &= \dfrac{e^{i\varphi}}{(2n+1)\sin \vartheta} \left[ (n+m)(n+m-1)Y^{m-1}_{n-1}(\vartheta, \varphi) - (n-m+1)(n-m+2)Y^{m-1}_{n+1}(\vartheta, \varphi) \right] \label{kugel:eq:rec-sph_harm-4} + \end{align} +\end{subequations} -\section{Series Expansions in $L^2(S^2)$} +\begin{enumerate}[(i)] + \item + \begin{proof} + We can multiply both sides of equality in eq.\eqref{kugel:eq:rec-leg-1} by $e^{im \varphi}$ and perform the substitution $z=\cos \vartheta$. After a few simple algebraic steps, we will obtain the relation we are looking for + \end{proof} + \item + \begin{proof} + In this proof, as before, we can perform the substitution $z=\cos \vartheta$, and notice that $\sqrt{1-z^2}=\sin \vartheta$, hence, the relation in eq.\eqref{kugel:eq:rec-leg-2} will be + \begin{equation*} + \frac{2m \cos \vartheta}{\sin \vartheta} P^m_n(\cos \vartheta) = P^{m+1}_n(\cos \vartheta) + [n(n+1)-m(m-1)]P^{m-1}_n P^m_n(\cos \vartheta). + \end{equation*} + The latter, multiplied by $e^{im\varphi}$, becomes + \begin{align*} + \frac{2m \cos \vartheta}{\sin \vartheta} P^m_n(\cos \vartheta)e^{im\varphi} &= P^{m+1}_n(\cos \vartheta)e^{im\varphi} + [n(n+1)-m(m-1)]P^{m-1}_n P^m_n(\cos \vartheta)e^{im\varphi} \\ + &= P^{m+1}_n(\cos \vartheta)e^{i(m+1)\varphi}e^{-i\varphi} + [n(n+1)-m(m-1)]P^{m-1}_n (\cos \vartheta)e^{i(m-1)\varphi}e^{i\varphi} \\ + &= Y^{m+1}_n(\vartheta, \varphi)e^{-i\varphi} + [n(n+1)-m(m-1)]Y^{m-1}_n(\vartheta, \varphi)e^{i\varphi}. + \end{align*} + Finally, after some ``cleaning'' + \begin{equation*} + Y^m_n(\vartheta, \varphi) = \frac{\tan \vartheta}{2m} \left[ Y^{m+1}_n(\vartheta, \varphi)e^{-i\varphi} + [n(n+1)-m(m-1)]Y^{m-1}_n(\vartheta, \varphi)e^{i\varphi} \right] + \end{equation*} + \end{proof} + \item + \begin{proof} + Now we can consider eq.\eqref{kugel:eq:rec-leg-3}, and multiply it by $e^{im\varphi}$. After the usual substitution $z=\cos \vartheta$, we have + \begin{align*} + \sin \vartheta P^m_n(\cos \vartheta)e^{im\varphi} &= \dfrac{e^{im\varphi}}{2n+1}\left[ P^{m+1}_{n+1}(\cos \vartheta) - P^{m+1}_{n-1}(\cos \vartheta)\right] \\ + &= \dfrac{e^{-i\varphi}}{2n+1}\left[ P^{m+1}_{n+1}(\cos \vartheta)e^{i(m+1)\varphi} - P^{m+1}_{n-1}(\cos \vartheta)e^{i(m+1)\varphi}\right]. + \end{align*} + A few manipulations later, we will obtain + \begin{equation*} + Y^m_n(\vartheta, \varphi) = \frac{e^{-i\varphi}}{(2n+1)\sin \vartheta} \left[ Y^{m+1}_{n+1}(\vartheta, \varphi)-Y^{m+1}_{n-1}(\vartheta, \varphi) \right]. + \end{equation*} + \end{proof} + \item + \begin{proof} + This proof is very similar to the previous one. We just have to perform the substitution $z = \cos \vartheta$, as always. Secondly we can multiply the right side by $e^{im\varphi}$ and the left one too but in a different form, namely $e^{im\varphi}=e^{i(m-1)\varphi}e^{i\varphi}$. Then it is only a question of recalling the definition of $Y^m_n(\vartheta, \varphi)$. + \end{proof} +\end{enumerate} -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} +\section{Series Expansions in $L^2(S^2)$} +We have now reach a point where we have all the tools that are necessary to build something truly amazing: a general series expansion formula for +function on the surface of the sphere. +Before starting we want to recall the definition of the inner product on the spherical surface 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 + \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. +To be a bit technical we can say that the set of spherical harmonic functions, toghether with the inner product just showed, +form something that we call Hilbert Space\footnote{For more details about Hilber space you can take a look in section \ref{kugel:sec:preliminaries}}. +This function space is defined over the space of ``well-behaved'' \footnote{The definitions of ``well-behaved'' is pretty ambigous, even for mathematicians. +It depends basically on the context. +You can sumarize it by saying: functions for which the theory we are considering (Fourier theorem) is always true. In our case we can say that well-behaved functions +are functions that follow some convergence contraints (pointwise, uniform, absolute, ...) that we don't want to consider further anyway.} 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. + +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{}. \subsection{Spherical Harmonics Series} @@ -714,11 +882,96 @@ 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} + \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} + \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: +\begin{equation*} + 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 $n$. + +\begin{lemma}[Spherical harmonic coefficients] + \label{kugel:lemma:spherical-harmonic-coefficient} + \begin{align*} + 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*} +\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}: + \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'} + \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} + \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 +\begin{theorem}[Fourier Theorem on $\partial S$] + \label{fourier-theorem-spherical-surface} \begin{equation*} - \hat{f}(\vartheta, \varphi) - = \sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}} - c_{m,n} Y^m_n(\vartheta, \varphi) + \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{definition} +\end{theorem} +The connection to Theorem \ref{fourier-theorem-1D} is pretty obvious. + +\subsection{Spectrum} + +\begin{figure} + \centering + \kugelplaceholderfig{.8\textwidth}{5cm} + \caption{\kugeltodo{Rectangular signal and his spectrum.}} + \label{kugel:fig:1d-fourier} +\end{figure} + +In the case of the classical one-dimensional Fourier theory, we call \emph{Spectrum} the relation between the fourier coefficients $c_n$ and the multiple +of the fundamental frequency $2\pi/T$, namely $n 2\pi/T$. In the most general case $c_n$ are complex numbers, so we divide the concept of spectrum in +\emph{Amplitude Spectrum} and \emph{Phase Spectrum}. In fig.\ref{kugel:fig:1d-fourier} a function $f(x)$ is presented along with the amplitude spectrum. + +\begin{figure} + \centering + \kugelplaceholderfig{.8\textwidth}{7cm} + \caption{\kugeltodo{Confront between image reconstructed only with phase and one only with amplitued}} + \label{kugel:fig:phase&litude-2d-fourier} +\end{figure} + +The thing that is easiest for us humans to visualize and understand is often the Amplitude Spectrum. +This is a huge limitation, since for example in Image Processing can be showed in a nice way that much more information is contained in the phase part (see fig.\ref{kugel:fig:phase-2d-fourier}). + +\begin{figure} + \centering + \kugelplaceholderfig{.8\textwidth}{9cm} + \caption{\kugeltodo{fig that show fourier style reconstruction on sphere (with increasing index)}} + \label{kugel:fig:fourier-on-sphere-increasing-index} +\end{figure} + +The same logic can be extended to the spherical harmonic coefficients $c_{m,n}$. In fig.\ref{kugel:fig:fourier-on-sphere-increasing-index} you can see the same concept as in fig.\ref{kugel:fig:1d-fourier} +but with a spherical function $f(\vartheta, \varphi)$. + +\subsection{Energy of a function $f(\vartheta, \varpi)$} + +\begin{lemma}[Energy of a spherical function (\emph{Parseval's theorem})] + \begin{equation*} + \int_0^{2\pi}\int_0^\pi |f(\vartheta, \varphi)|^2 \sin\vartheta \, d\varphi \, d\varphi = \sum_{n=0}^\infty \frac{1}{2n+1} \sum_{m=-n}^n |c_{m,n}|^2. + \end{equation*} +\end{lemma} +\begin{proof} +\end{proof} -\subsection{Fourier on $S^2$} +\subsection{Visualization}
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