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Just maths!


Spherical Harmonics

Spherical harmonics can be generalized in the same way as Fourier series, simply in one dimension higher. Motivation for considering spherical harmonics is the same as for Fourier series, e.g., both diagonalize all linear operators that commute with rotations. Spherical harmonics are eigenfunctions of the sphericalLaplacian.

Preliminaries

The $L^{2}$ inner product of the two functions $f$ and $g$ is given by
\begin{align}
\langle f, g \rangle = \int f(s)\bar{g}(s)ds,
\end{align}
where $\bar{g}$ is the complex conjugate and the integral is over the space of interest, for example over the circle, $\int_{0}^{2\pi}d\theta$, or over the sphere,
$\int_{0}^{2\pi}\int_{0}^{\pi}{\rm sin}\phi d\phi d\theta$.

We define the $L^{2}$ norm of a function $f$ by $||f|| = \sqrt{\langle f, g \rangle}.$

The space $L^{2}$ norm consists of all functions, $f$ such that $||f|| < \infty$.

A basis for $L^{2}$ space is a set of all functions ${\psi}$ with the properties:

1. Orthogonal: $\langle \psi_{i}, \psi_{j}\rangle = 0, \ {\rm for } \ i \ne j$.
2. Normalized: $||\psi_{i}|| = 1$.
3. Spanning: Any function in $L^{2}$ can be written as linear combination of $\psi_{i}'s$; $f = \sum_{i}\alpha_{i} \psi_{i}$, where $\alpha_{i}$ are complex numbers.

The notation $f(x) = \mathcal{O}(g(x))$ means ${\rm lim}(|f(x)/g(x)|)<C<\infty$, where the limit is towards the point of interest, usually $\infty$.


Fourier Series


Spherical Harmonics can be generated in the same way as Fourier Series, simply in one dimension higher. One cannot understand Spherical Harmonics without first understanding Fourier series. Conversely, a good understanding of Fourier series will get you much of the way through the analysis of Spherical Harmonics.

We start with something familiar, polynomials in $\mathbb{R}^{2}, \ \{(x,y): x, y \in \mathbb{R} \}$. Then we restrict our attention
to those polynomials that are \textit{harmonic}, meaning $\triangle_{2}p(x,y) = 0$, where $\triangle_{2}$ is the two-dimensional \textit{Laplacian},

\begin{align}\label{laplacian2}
\triangle_{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}.
\end{align}

An example of harmonic polynomial is $\triangle_{2}(x + x^{2} - y^{2}) = 0$.

Homogeneous harmonic polynomials

Next we further restrict our consideration to homogeneous harmonic polynomials.


Definition: A function $p(x,y)$ is homogeneous of degree $n$ if $p(tx, ty) = t^{n}p(x,y)$ for all positive real numbers $t$.


Examples of homogeneous harmonic polynomials of degrees $0, \ 1, \ 2,$ and $3$ are $6$, $x$, $x^{2} - y^{2}$, and $3x^{2}y - y^{3}$.

Homogeneous functions are nicely represented in polar coordinates $\{(r, \theta): r\in \mathbb{R}_{+}, \theta \in [0, 2\pi] \}$, where $(x,y) = (r{\rm cos}\theta, r{\rm sin}\theta)$.

In polar coordinates, a function $p_{n}(x,y)$ that is homogeneous of degree $n$ can be written as $p_{n}(x,y) = r^{n}q_{n}(\theta)$ for some function $q_{n}$.

In polar coordinates, the Laplacian (\ref{laplacian2}) becomes

\begin{align}
\triangle_{2} = \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2} + \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r}.
\end{align}

A homogeneous harmonic polynomial of degree $n \ (n\ge 0)$ satisfies

\begin{equation}\label{laplace}
\begin{split}
0 &\quad = \triangle_{2}p_{n}(x,y) \\
&\quad = n(n-1)r^{n-2}q_{n}(\theta) + \frac{1}{r}nr^{n-1}q_{n}(\theta) + \frac{1}{r^2}r^{n}q_{n}^{''}(\theta)\\
&\quad = r^{n-2}\left(n^2q_{n}(\theta) + q_{n}^{''}(\theta) \right)
\end{split}
\end{equation}

The operator $\triangle_{2}$ acts on homogeneous functions in a simple way, separately in $r$ and $\theta$. In $r$ it simply reduces the power by two.
In $\theta$ it acts using the restriction of $\triangle_{2}$ to the circle, i.e. the \textit{Circular Laplacian}, $\triangle_{2S} = \frac{\partial^2}{\partial \theta^2}$.

For (\ref{laplace}) to hold, $q_n$ must be an eigenfunction of $\triangle_{2S}$, i.e. $\triangle_{2S}q_n = -n^{2}q_{n}$. The operator $\triangle_{2S}$ is self-adjoint on the circle, meaning

\begin{align}
\langle \triangle_{2S}f(\theta), g(\theta) \rangle = \langle f(\theta), \triangle_{2S}g(\theta) \rangle.
\end{align}

This fact follows by integration by parts. Self-adjoint operators have the property that eigenfunctions with different eigenvalues are orthogonal.

We have

\begin{equation}
\begin{split}
-n^{2}\langle q_{n}(\theta), q_{m}(\theta) \rangle &\quad = \langle \triangle_{2S}q_{n}(\theta), q_{m}(\theta) \rangle = \langle q_{n}(\theta), \triangle_{2S}q_{m}(\theta) \rangle \\
&\quad = -m^{2}\langle q_{n}(\theta), q_{m}(\theta) \rangle,
\end{split}
\end{equation}

so if $n \ne m$, we must have $\langle q_{n}(\theta), q_{m}(\theta) \rangle = 0$.

The next question is whether there can be more than one $q_{n}$ for the same $n$. The $q_{n}$ satisfies $q^{''}_{n}(\theta) = -n^{2}q(\theta)$. All solutions of this differential equation
are of the form
\begin{align}
q_{n}(\theta) = A{\rm cos}(n\theta) + B{\rm sin}(n\theta) \ {\rm or}
\end{align}

equivalently

\begin{align}
q_{n}(\theta) = Ce^{in\theta} + De^{-in\theta}.
\end{align}

Thus there are many possibilities for $q_{n}(\theta)$, but we can generate them all as linear combinations of two elements, $\{ e^{in\theta}, e^{-in\theta} \}$, chosen to be orthogonal. The eigenspaces are
thus two dimensional (for $n =0$ it is only one dimensional). This construction, now complete, has generated the Fourier series.

Let's consider next what we have given up by using only homogeneous, harmonic polynomials.

Theorem: Any polynomial of degree $n$, when restricted to the circle, can be written as a sum of homogeneous harmonic polynomials of degree at most $n$.

On the circle itself, we lose nothing by our restriction. Again, restricted to the circle, polynomials can approximate any continuous function, and thus any $L^{2}$ function, to any desired degree of accuracy. Polynomials are dense in $L^{2}(S^{1})$. The exponentials $\{e^{in\theta} \}$, $n\in \mathbb{Z}$ thus span $L^{2}$. After normalization, they will be a basis. There are of course independent proofs that the Fourier series are a basis for $L^{2}(S^{1})$.

Up to this point, we have constructed the Fourier series basis for the cirlce by:

1. restricting $\triangle_{2}$ to the circle,
2. decomposing into the eigenspaces of $\triangle_{2S}$, and then
3. taking a convenient basis within each eigenspace.

$\mathbb{R}^{3}$: Ordinary Spherical Harmonics

The construction under Spherical Harmonics section can be done in any dimension. The sphere $S^{1}$ in $\mathbb{R}^{2}$ is normally called the
circle, but we could equally well call it a sphere and say the Fourier Series are Spherical Harmonics. The usual usage for Spherical Harmonics refers to the Surface Spherical Harmonics on the sphere $S^{2}$ in $\mathbb{R}^{3}$ . We shall follow this usage and examine this case in this section.

In $\mathbb{R}^{3}$, $\{(x,y,z): x,y,z \in \mathbb{R}\}$, the Laplacian is

\begin{align}
\triangle_{3} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}.
\end{align}

In spherical cordinates $\{(r, \theta, \phi): r\in \mathbb{R}_{+}, \theta \in [0, 2\pi), \phi \in [0, \pi] \}$ where

$(x, y, z) = (r{\rm cos}\theta {\rm sin}\phi, r{\rm sin}\theta {\rm sin}\phi, r{\rm cos}\phi)$,

\begin{align}
\triangle_{3} = {\rm csc}^{2}\phi \frac{\partial^{2}}{\partial \theta^{2}} + \frac{\partial^{2}}{\partial \phi^{2}} + {\rm cot}\phi \frac{\partial}{\partial \phi} + \partial r \frac{\partial}{\partial r} + r^{2} \frac{\partial^{2}}{\partial r^{2}}.
\end{align}

Suppose, as above, we have a homogeneous harmonic polynomial of degree $n$, $p_{n}(x,y,z) = r^{n}q_{n}(\theta, \phi)$. We then have

\begin{align}\label{lapl3S}
\begin{split}
0 &\quad = \triangle_{3}p_{n} = \left[{\rm csc}^{2}\phi \frac{\partial^{2}}{\partial \theta^{2}} + \frac{\partial^{2}}{\partial \phi^{2}} + {\rm cot}\phi \frac{\partial}{\partial \phi} \right] r^{n}q_{n}(\theta, \phi)\\
&\quad + 2rnr^{n-1}q_{n}(\theta, \phi) + r^{2}n(n-1)r^{n-2}q_{n}(\theta, \phi)\\
&\quad = r^{n}\left(\left[{\rm csc}^{2}\phi \frac{\partial^{2}}{\partial \theta^{2}} + \frac{\partial^{2}}{\partial \phi^{2}} + {\rm cot}\phi \frac{\partial}{\partial \phi} \right] +n(n-1)\right)q_{n}(\theta, \phi).
\end{split}
\end{align}

Defining the spherical Laplacian by

\begin{align}
\triangle_{3S} = {\rm csc}^{2}\phi \frac{\partial^{2}}{\partial \theta^{2}} + \frac{\partial^{2}}{\partial \phi^{2}} + {\rm cot}\phi \frac{\partial}{\partial \phi},
\end{align}

from equation (\ref{lapl3S}) we have

\begin{align}
\triangle_{3S}q_{n}(\theta, \phi) = -n(n+1)q_{n}(\theta, \phi).
\end{align}

This $q_{n}$ is therefore an eigenfunction of the Spherical Laplacian. Any such eigenfunction is called a Spherical Harmonic.

As before, we note that $\triangle_{3S}$ is self-adjoint, which implies that the eigenspaces $\Lambda_{n}$ are orthogonal. The space $\Lambda_{n}$ consists of the homogeneous harmonic polynomials of degree $n$ restricted to the sphere, and has dimension $2n + 1$. On the sphere, the homogeneous harmonic polynomials span the set of all polynomials, which in turn are dense in $L^{2}$. Our spherical harmonics therefore span $L^{2}$. If we take a basis within each eigenspace then this collection will give a basis for $L^{2}$ of the sphere. Thus Spherical Harmonics arise in $\mathbb{R}^{3}$ in the same way Fourier series arise in $\mathbb{R}^{2}$. They are consequently sometimes called 'Fourier series on the
Sphere'.

For a fixed $n$, we can organize a basis for $\Lambda_{n}$ as

\begin{align}
\left \{\frac{e^{im\theta}}{\sqrt{2\pi}} P_{n}^{m}({\rm cos}\phi) \right \}_{-n \le m \le n}
\end{align}

Source: MartinJ. Mohlenkamp.

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Prof. Joyce Ndalichako, the Minister of Education, Science and Technology delivers speech on 7th May 2021 during Science, Technology, and Innovation competition finals opening ceremony at Jamhuri stadium in Dodoma:

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