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Questions tagged [spherical-harmonics]

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

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Simulate a sample function with known power spectrum

There is a known power spectrum $$C_l=\frac1{2l+1}\sum_{m=-l}^l |a_{lm}|^2$$ for some $l=0,1,\dots,l_{max}$ where $$ a_{lm}=\int_{-\pi/2}^{\pi/2} \mathrm \int_0^{2\pi}f(\theta,\phi)Y^*_{lm}(\theta,\...
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Expressing spherical functions with zonal harmonics

$\newcommand{\d}{\mathrm d}$First time using StackExchange, so please excuse any mistakes: My goal is to express functions on the sphere as a sum of zonal harmonics. I've spent a while with the ...
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Is this minimisation problem convex and how can this be (dis)proved?

Is the following minimisation problem convex? $$\min\limits_{\phi_1,\phi_2,\phi_3}\left(~\sum\limits_{l=1}^{l_{\mathrm{max}}} ~\sum\limits_{ m=0}^{l}~\left|(\hat{f}_{lm})^{g} -\hat{f}_{lm} \right| ^2 ...
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Proof for a Legendre polynomial identity on a 2-sphere that appears in cosmology

How can I prove the following identity? \begin{equation} \int d\Omega_{\hat k} P_l(\hat n \cdot \hat k) P_{l'}(\hat n'\cdot \hat k)=\frac{4\pi}{2l+1} P_l(\hat n\cdot \hat n')\delta_{ll'}\label{PP} \...
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Understanding the Following Integral Notation

I'm a little confused on the notation my professor used for the following integral. \begin{equation} \int \bar{Y}_{l_f}^{m_f} \left( \dfrac{-Y_1^1 + Y_1^{-1}}{\sqrt{2}}, \dfrac{iY_1^1 + iY_1^{-1}}{\...
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Vector spherical harmonics and resolution

Consider the analysis of a vector function ${\bf F}(\theta,\phi)$ in terms of the vector spherical harmonic (VSH) basis ${\bf Y}_{jm}^{l}(\theta,\phi)$ where $l=j-1,j,j+1$, and $j=0,\ldots, \infty$ ...
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45 views

Is $a^4 + b^4 + c^4$ a spherical harmonic?

I am trying to construct Harmonic polynomials on the sphere. What about the examples: $f = x^2 + y^2 + z^2$ $g = x^4 + y^4 + z^4$ We have that $\nabla^2 f \neq 0$ by compuing the second derivative ...
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32 views

Spherical expansion of an exponential function?

We know that a normal planewave can be Rayleigh expanded by spherical harmonics as$$e^{i\vec k·\vec r}=4π\sum_{l=0}^∞\sum_{m=-l}^li^lj_l(kr)Y_{lm}(\hat{\vec k})Y_{lm}^*(\hat{\vec r}).$$ Does any body ...
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Does the Addition theorem still holds for real (tesseral) spherical harmonics?

the Addition Theorem state that $$P_{l}(\cos \gamma) = K_l\sum_{m=-l}^{+l} Y_{l}^{m}(\theta, \phi) ~\bar{Y}_{l}^{m}(\theta', \phi')$$ and i have that for real function spherical harminics are ...
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30 views

Expanding a function in spherical coordinates

I have a function f(theta,phi,r) in spherical coordinates. The function dies out at r->infinity (r in my case is dimensionless). Is there a natural way of expanding the function, the same way a ...
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83 views

on the limit of the finite representation of harmonics

Let $Y_n^j, \, -n\leq j \leq n$ be a spherical harmonic. Let $f$ be an even function on the sphere. Then, $f$ can be introduced as $$ f=\sum_{j=-n}^A\hat f(j)Y_n^j, $$ where $\sum_{j=-n}^A|a_j|>0$ ...
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$\Phi(\phi)=A e^{-im \phi}+B e^{im \phi}$ and $\Theta(\theta)=P_l^m(\cos \theta)$ are combined to $P_l^m(\cos \theta) e^{im \phi}$?

$\Phi(\phi)=A e^{-im \phi}+B e^{im \phi}$ and $\Theta(\theta)=P_l^m(\cos \theta)$ are combined to form $$P_l^m(\cos \theta) e^{im \phi}$$ How? Where did $A,B, e^{-im \phi}$ go? Does it read ...
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Boundary conditions and decomposition on spherical harmonics

Say I have a function $f(x,y,z)$ satisfying some Laplace-type equation on a domain. This domain is an hemisphere with boundary defined by $x=0$ in Cartesian coordinates. I impose either Dirichlet or ...
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Integration of spherical harmonics

for my research i found myself in place to compute the following integral involving the product of spherical harmonics, $Y_l^m(\theta,\phi)$, and gradient of spherical harmonics. $\int_R^{\infty} \...
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38 views

Spherical harmonics filtering

Consider a signal $U$ defined on the 2-sphere that is expressed as the product of two functions $A,B$, or $$ \begin{aligned} U(\theta,\phi) &= \sum_{n=0}^{\infty}\sum_{m=-n}^{n}u_{nm}Y_n^m(\theta,\...
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Property of polynomials associated with mirror related representations

Let $\mathcal{P}$ be point symmetry group with $\Gamma$ and $\Gamma'$ being two irreducible representations, that merely differ in the sign of the character of the mirrored sector. Also, let $f_\Gamma$...
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Regarding the dimension of irreducible (finite-dimensional) group representations

Ok, I admit it. I'm confused. I'm a physics student attempting to learn some group theory and topology in my spare time. I was reading about group representations. For example I get that the set of ...
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84 views

Does this oscillatory integral exist?

Let $n\geq 2$ and consider the improper integral $$I:=\int_{\mathbb{R}^{n}}F(x)dx$$ where $F$ is a continuous function. If $I$ exists then $$I=\lim_{R\rightarrow +\infty}\int_{B_{R}}F(x)dx,$$ ...
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Spherical harmonics of a non-negative function of the two sphere

I am working on a data analysis project, as a part of which I want to express a probability distribution as a spherical harmonic expansion on a 2-sphere. Imposing the condition of realness of the ...
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28 views

Equivalence of spherical harmonic definitions

In Griffith's "Introduction to quantum mechanics", the spherical harmonics are defined for integers $l$ and $-l\leq m\leq l$ as $$Y_{l}^m(\theta,\phi) = \epsilon \sqrt{\frac{2l+1}{4\pi}\frac{(l-|m|)!}...
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Definite integration of spherical Bessel function of radical argument

I have to solve some integrals of the form: $$\int_0^{x_0} dx \, j_n( R ) \cdot \frac{p(x)}{R^n}$$ where $R=\sqrt{x^2 + 2 a c x + c^2}$ , $j_n$ is the spherical Bessel function of order n, p(x) is a ...
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Can we prove that hex grids are better for spherical harmonics than cartesian grids are?

In for example physics, spherical harmonics seem to be very interesting functions for a multitude of reasons. Is there some way to show that if sampling on a hexagonal grid, we will be able to ...
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58 views

Decomposition of polynomial space restricted to unit sphere into harmonic subspaces

Let $S^{d-1}$ be the unit sphere in $\mathbb{R}^d$ and $\operatorname{Pol}_{\leq n}(S^{d-1})$ be the space of polynomial functions of degree at most $n$. I'm trying to understand the irreducible ...
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145 views

Spherical harmonics and Dirac delta integrals

I have an equation given in the following form: $$D_{l,m} = \int_{\mathbf{\Omega}}^{} \mathrm{d}\Omega \ Y_{l,m}(\mathbf{\hat{s}})\int_{\mathbf{\Omega'}}^{} \mathrm{d}\Omega' \ K(\mathbf{\hat{s}} \...
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136 views

Steady State Temperature Distribution in Unbounded Region (Laplace's Equation in Spherical Coordinates)

I'm trying to find the steady state temperature distribution in the infinite region outside a sphere of unit radius centred on the origin, where the temperature takes the value $u = f(\theta$) on the ...
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37 views

Quantum Mechanics - Orthonormal Basis integration and Kronecker delta

Given that this integral I'm trying to solve is $$\frac{2}{\pi}\sum^{\infty}_{l=0}\sum^{l}_{m=-l}\int_{r=0}^{\infty}\int_{k=0}^{\infty} R_{nl}(r)b_{lm}(k)j_{l}(kr)k^2 r^2 \int_{\theta = 0}^{\pi}\int_{\...
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Irrep. of SU(2) and Laplace Eigenspaces

In order to calculate the Dirac spectrum on Berger's sphere $(S^3,g_t)$, I came across irreps of SU(2) (see Hitchin p. 30). Apperently, Hitchin restricts the Dirac operator to the eigenspaces of the ...
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1answer
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What am I doing wrong in this integration (over the angle between any two three-dimensional vectors)?

I am trying to do a six dimensional integration involving functions of two vectors $\vec{p}$ and $\vec{p}'$ in spherical coordinates. So $\vec{p}$ is reprsented by $\{p,\theta,\phi\}$, and $\vec{p}'$ ...
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Earth gravitational potential

In the Cook's "Perturbations of satellite orbits by tesseral harmonics" i've found the following formula(13, page 799) for potential: $$U = \frac{\mu}{r}\left\{1-\sum\limits_{n=2}^{\infty}\left(\frac{...
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What is an Orientation Distribution Function (ODF) and what does it represent in diffusion-MRI?

I am reading up on Diffusion MRI where there is discussion on Orientation Distribution Function (ODF), to represent the angular aspect of diffusion probability by a spherical ODF. Can anyone please ...
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eigenfunctions and eigenvalues of multiplication by spherical harmonic

Let $Y_{\ell m}$ be a real spherical harmonic, and define an operator on functions $f:S^2\to\mathbb{R}$ by $$(L_{\ell m}f)(\theta,\phi) = Y_{\ell m}(\theta,\phi)f(\theta,\phi).$$ What are the ...
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60 views

Prove that $\mathbb P(\|\nabla f(x)\|< Bn) < KB^2$ for a random spherical harmonic $f$

I have a random spherical harmonic of degree $n$ on the sphere $S^2$, i.e. $$ f = \sum_{k=-n}^{n} \xi_k Y_k$$ with $\xi_k \sim N\left(0, \dfrac{1}{2n+1}\right)$ being independent Gaussians and $\{Y_k\}...
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Simplifying with spherical harmonics

I originally asked this on the physics Stack Exchange site, but perhaps it could be more easily answered here. Given the definition of the correlation function for CMB temperature fluctuations as $$ ...
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128 views

Find the steady-state temperature distribution using spherical harmonics

I'm trying to find the steady-state temperature distribution inside a sphere of radius 1 with the surface temperature given by $$u(1,\theta,\phi)=3 \sin(\theta) \cos(\theta) \sin(\phi)$$ So I ...
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Deriving a new set of complete orthogonal basis functions inside an interval?

I am quite a noob in this area, so bear with me if my question doesn't make sense, and be kind enough to let me know why it is so. I am interested in deriving a new set of orthogonal basis functions (...
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A question about orthogonal functions and Laplace's equation

Suppose we have a harmonic function in spherical coordinates, $$ \nabla^2 f(r,\theta,\phi) = 0 $$ By using the separation of variables method to solve the PDE, we will find $$ f(r,\theta,\phi) = \sum_{...
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Are spherical harmonics harmonic?

According to Wikipedia, a harmonic function is one which satisfies: $$ \nabla^2 f = 0 $$ The spherical harmonics (also according to Wikipedia) satisfy the relation $$ \nabla^2 Y_l^m(\theta,\phi) = -\...
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Diagonalizing the rotation operator on functions on the sphere

Using (the exponential form of) Fourier series, we can diagonalize the rotation operator $S_\theta$ that rotates complex-valued functions on $S^1$ by $\theta$ , $f(x) \mapsto f(x-\theta)$. Can we do ...
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Are there any complete orthogonal basis functions inside unit sphere?

I went through so much literature, but couldn't find any orthogonal complete functions within the boundary $0 < r <1, 0 < \theta <\pi, 0 < \phi <2\pi$ other than 3D Zernike ...
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Fourier transform of surface measure, did I get it right?

I am trying to understand the Fourier transform of the surface measure. Here is the definition: If $d\sigma$ is the surface measure on the sphere $\mathbb{S}^{n-1}$, then $$\widehat{d\sigma}(\xi):=\...
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Laplace equation on a sphere

I am trying to solve Laplace equation on a sphere of radius $1$. $$\Delta u =0, B(0,1) \subset \mathbb{R}^3 $$ $$u|_{S(0,1)}=z^2$$ Of course, I deduced through separation of variables and some ...
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Inequality for spherical harmonics

In this article, it is claimed that (Claim 2.2), for any spherical harmonic $f$ of degree $n$ and any point $x \in \mathbb{S}^2$, we have $$|f(x)|^2 \leq C_1 n^2 \int_{D(x, \frac{1}{n})}f^2, $$ $$|\...
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Derivatives of an expansion of a function

Recall that, for fixed $j \in Z_+\cup \{0\}$, a spherical harmonic of degree $j$ is the restriction to $S^{n-1}$ of a harmonic polynomial on $R^n$ that is homogeneous of degree $j$. Let $Y_{j,1}\, ,Y_{...
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Is there a way to simplify this quotient of factorials?

So I'm implementing numerically this quotient of factorials (in spherical harmonics) and after $\nu=170$ I'm getting an overload error and I was wondering if there was a way to simplify this ...
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Modeling Lambertian surface by using second order spherical harmonics lighting?

For a 3D surface assumed that its surface appearance follows the Lambertian reflectance i.e. $I = \rho (\mathbb{l} \cdot \mathbb{n})$, I found that it also can be expressed by using following second-...
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112 views

Calculating a normal vector field for a surface defined by spherical harmonics

Given a surface defined by spherical harmonic terms $f(\theta,\phi) = \sum_{l=0}^\infty\sum_{m=-l}^lf_l^mY_l^m(\theta,\phi)$ How can I evaluate the normal vector at each location on the surface? i....
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Are the dot products of all vector spherical harmonics complete?

Does the set of all dot products ${\bf Y}^j_{jm} (\theta, \phi) \cdot {\bf Y}^{j'}_{j'm'}(\theta,\phi)$, where ${\bf Y}^j_{jm}$ are vector spherical harmonics ($j,j' = 0,1,2, ...$ and $m,m' = -j, ..., ...
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Integral of product of three spherical harmonics with derivatives

In a previous question, the integral of products of three spherical harmonics was discussed: \begin{align}\int_0^{2\pi}\int_0^\pi Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\theta,\phi)&Y_{l_3}^{m_3}...
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394 views

Spherical Harmonic Derivative

This question is a follow up to a previous question: Spherical Harmonic Identity. Instead of using the above question's method, I tried something like this, but don't get the same result and I'm ...
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1answer
105 views

Spherical Harmonic Identity

I've been told there exists the identity $$\partial_\theta Y_{\ell,m}(\theta,\phi)=\frac{1}{2}e^{-i\phi}\sqrt{(\ell-m)(\ell+m+1)}Y_{\ell,m+1}-\frac{1}{2}e^{i\phi}\sqrt{(\ell+m)(\ell-m+1)}Y_{\ell,m-1},...