# Tagged Questions

Questions on spherical harmonics, a set of basis functions that satisfy an orthogonality relation over the sphere.

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### Spherical Harmonics Interpolation

I am interested in interpolation using spherical harmonics. I feel like I searched all Google pages containing key words of this subject. Therefore I wanted to ask I if anyone is familiar to this and ...
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### Expressing a function as a sum of spherical harmonics

I'm asked to express the function $$f(\theta,\phi) = \sin\theta \left[ \sin^2\frac{\theta}{2}\,\cos\phi + i \cos^2\frac{\theta}{2}\,\sin\phi \right] + \sin^2\frac{\theta}{2}$$ as a sum of spherical-...
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### $H^k= \{f \in P^k : \Delta f = 0\}$?

Let $P^k=$homogeneous polynomials of degree $k$ in $x$, $y$, $z$, $k=0, 1, 2, \dots,$, i.e. $P^k= \text{Span} \{x^{k_x}y^{k_y}z^{k_z} : k_x+k_y+k_z=k\}$. This is maybe a silly question, but I am not ...
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### Spherical function dot product after rotation

Let's start from a system $f_n(x) = e^{inx}$. I can rotate one element: $f_m(x + x_0) = e^{imx_0} e^{imx}$ , which is still orthogonal to all but one element of system. Same holds for $n$-...
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### When expanding a function in a sphere, why is the complex conjugate of the spherical harmonic function used to calculate the coefficients?

When expanding a function on a sphere $f(r,\theta, \phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}=A_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta,\phi)$. Since what I'm asking involves the ...
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### Spherical functions which are invariant under a finite rotation group

Is there a nice, clean reference which lists a basis in terms of (linear combinations of) spherical harmonics for the $L^2$ space of functions defined on the sphere $\mathbb{S}^2$ which are invariant ...
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### Rotationally invariant function in three-sphere

I know that every rotationally invariant function $f(x,y,z)$ in two-sphere $S^2$ must satisfy \begin{align*} f(x,y,z)=f(h(\theta)(x,y,z)^T) \end{align*} for all $\theta$ and $x,y,z$, where \begin{...
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### Spherical Harmonic integration

How do I solve the lower integration of spherical harmonics, in terms of Clebsch Gordon coefficients? \int d\Omega Y^{l_1}_{m_1}(\hat n)Y^{l_2}_{m_2}(\hat n){Y^{l}_m}^*(\hat n) \end{...
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### Equality about integral on unit sphere, involving Lebesgue measure

I met an equality in the book "Harmonic analysis and approximation on the unit sphere" by Wang Kunyang, in page 23. I cannot follow the reasoning below, \begin{align} \int_{\Omega_{n}} P^{n}_{k}(\xi\...
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### Proof of an inequality in spherical harmonics using Minkowski's inequality

I met an inequality in the book "Harmonic analysis and approximation on the unit sphere" by Wang Kunyang and Li Luoqing. I need some hints to follow the proof. The proof is following, \begin{align} \...
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### An expository reference on spherical harmonics on $S^n$.

I am looking for a thorough reference which explains how to compute the spherical harmonics on $S^n$ and how to upper and lower-bound their values. About the first part of my query, I am not so much ...
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### Spherical Harmonic fitting excess polar magnitudes

I am trying to fit an expansion of spherical harmonic functions to a dataset distributed over the surface of a sphere using the least squares method. Each data point is in terms of (r,Î¸,Ï†) where r is ...
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### How to prove the given property of spherical harmonics?

How to prove: $$\int^\pi_0\int^{2\pi}_0 Y_{l'',m''_l}(\theta,\phi) Y_{l',m'_l}(\theta,\phi) Y_{l,m_l}(\theta,\phi) \sin\theta \,d\theta \,d\phi = 0$$ unless $l, l',$ and $l''$ are integers that can ...
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### What do these variables mean in regard to the wave equation and spherical waves?

https://en.wikipedia.org/wiki/Wave_equation#Spherical_waves Before it states ''where K=w/c'', there is an equation that has the following variables: d,r,w,c,l. It also has f_lm(r) What do each of ...
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### Orthogonal functions and different function arguments

This should be a fairly trivial question, but I would like to support my current intuition. Let's consider two orthogonal functions $f$ and $g$, i.e. two function whose inner product is equal to zero ...
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### Closed-form of spherical expansion of Legendre polynomial $P_k(\sin{\theta}\cos{\varphi})$

During the times of working on some problem in astro/geophysics I have come across a problem involving an expansion into spherical harmonic functions (this is the remnance of nomenclature there used ...
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### Spherical Harmonics's sumation identity proof

Could anybody help me proving this identity? I've tried using the completeness property of the spherical harmonics, expanding the right side using the Binomial theorem and the left side as a ...