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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8 views

Function that takes Wigner D matrix and complex vector and maps to real valued rotated vector

I am looking for a function $f$, that takes a the product of a Wigner D matrix $D_{l=1}(\boldsymbol{R})$ and a complex vector $\boldsymbol{c} \in \mathbb{C}^3$ as input, and outputs a rotated real ...
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How to prove spherical harmonic addition theorem

I have been trying to prove that $$ P_\ell(\cos\gamma)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^\ell Y_{\ell m}^{*}(\theta', \phi')Y_{\ell m}(\theta, \phi) $$ for $\cos\gamma=\cos\theta\cos\theta'+\sin\...
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What is the canonical orthonormal basis of the space $L^2(\mathbb{S}^2)$

I'm going to express a function in Fourier series. From Fourier Analysis we know that $L^2(\mathbb{S}^1)=\overline{\bigoplus_{n\in \mathbb{Z}} \langle e^{in\theta}\rangle}$ , and for $L^2(\mathbb{S}^3)...
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65 views

Reference request on spherical harmonics

I'd like to find a (hopefully modern, mathematician-friendly) reference which proves that homogeneous harmonic polynomials restrict to an orthonormal basis for $L^2$ functions on the sphere $S^n$. ...
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132 views

Is every harmonic polynomial a linear combination of these?

In $N$-dimensional space, we can show by direct calculation that the polynomial $$ r^{2K+N-2}\nabla_a\nabla_b\nabla_c\cdots \frac{1}{r^{N-2}} \hspace{1cm} \text{(with $K$ derivatives)} $$ is ...
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172 views

Solid harmonic addition theorem in higher dimensions?

The solid harmonics are solutions to Laplace's equation in spherical coordinates. The regular and irregular solid harmonics, obtained by rescaling spherical harmonics, are respectively $$R_l^m(\textbf{...
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85 views

Spherical Harmonics Sum Identity

I'm taking a course in Quantum Mechanics and this problem is causing me some struggles. Can someone help me prove this identity? $$\sum_{m = -l}^l m^2 |Y_{l}^{m}(\theta, \phi)|^2 = \frac{l(l+1)(2l+1)}{...
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43 views

Localization property of spherical harmonics on incomplete spheres

This question follows from another question I asked here. I am currently trying to read this paper and I am having difficulty in understanding the interpretations of eq. 7 which are given in lines ...
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62 views

Spherical Harmonics on incomplete sphere [closed]

Let me start by saying that I am starting to fall in love with the spherical harmonics and analysis of functions defined on a sphere. I am a physicist studying Cosmology, so you can imagine I get to ...
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44 views

Cartesian Coordinates vs Spherical Coordinates vs Spherical Basis

I would like to properly understand spherical coordinates once and for all: In the years of innocence and youth we are all introduced to cartesian coordinates. This are pretty simple to understand: ...
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Are Spherical Harmonics of the same $l$ orthogonal on circular trajectories?

A well known property of spherical harmonics is their orthogonality, which allows for nice decomposition of functions on the unit sphere. The orthogonality of spherical harmonics takes the form of $$\...
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90 views

Parity of spherical harmonics

I would like to proof $Y_{\ell m}(-\mathbf{r}) = (-1)^\ell\, Y_{\ell m}(\mathbf{r})$. In this formula, $Y_{\ell m}$ are the spherical harmonics given by \begin{equation} Y_{\ell m}(\theta, \varphi) = \...
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39 views

Mercer's theorem for zonal kernel

Let $f$ be a continuous function $[-1,1]\to\mathbb{R}$. Consider an integral operator $A$ on the unit sphere $S^{d-1}$ of $\mathbb{R}^d$, which acts on $\phi\in\mathcal{L}^2(S^d)$ as $$A\,\phi(x) = \...
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106 views

Product of $\cos\theta$ or $\sin\theta$ and a Spherical Harmonic

To mix an orientation vector in spherical coordinates with a Spherical Harmonic, I am trying to find an expression for the product of $\cos\phi\sin\theta$, $\sin\phi\sin\theta$ or $\cos\theta$ with a ...
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energy of sparsely sampled spherical function

if I should estimate the "Energy" of a spherical function of which I have some sparsely sampled values, is it correct to simply sum the squares of my sample values, multiply by 4*PI and ...
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noisy, sparsely sampled spherical harmonics transform

I have samples of many spherical functions. There are between 200 and 1000 samples for each function, they are randomly distributed (with local clusters and void areas). I'm trying to do spherical ...
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22 views

Del of spherical harmonics

I am looking for the del of spherical harmonics. For the laplace operator we have the defining relation: $\Delta Y^l_m = l(l+1) Y^l_m$. So, what is with: $\nabla Y^l_m $
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Real Spherical Harmonics Expansion Coefficients

I am interested in approximating a real function $f(x)$ on a sphere with a series expansion of spherical harmonics $Y_{l}^{m} (x)$: $$f(x) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} c_l^m Y_l^m (x),$$ ...
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Is there an equivalent for closure relation for spherical Bessel function for the 6th power of $x$?

I am aware of the the closure relation for spherical Bessel functions $$ \int_0^\infty{dx\; x^2 \; j_l(k_1x)j_l(k_2x)} = \frac{\pi}{2k^2} \delta(k_1 - k_2) $$ as well as the orthogonality relation. I ...
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15 views

cartesian representation of hemispherical harmonics

I'm searching for the cartesian representations of Hemispherical Harmonics. So far I only found the order 0 and order 1 representations in this paper Ultimately I'm looking for a polinomial recurrence ...
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finding spherical harmonics coefficients for non-uniformly sampled spherical function

I have sampled spherical functions for which I want to compute spherical harmonics coefficients. Reading through https://pdfs.semanticscholar.org/83d9/28031e78f15d9813061b53d25a4e0274c751.pdf?_ga=2....
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Is it possible to ‚blur, a spherical harmonic analytically in Z direction?

I have spherical harmonics that I want to ‚blur‘ in Z-axis direction. One way of doing this would be to evaluate the SH multiple times at offset locations and average the results - but I wonder if ...
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Proper definition of “order” of spherical harmonics?

I'm looking into spherical harmonics, and the term 'order' seems to be used quite inconsistently. In google's spherical harmonics framework (https://github.com/google/spherical-harmonics), the ...
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42 views

How prove $\int_{S^{n-1}}f(x\cdot \omega)\,d\omega=\int_{S^{n-1}}f(-x\cdot \omega)\,d\omega$

Let $f:\mathbb{R}^{n}\rightarrow \mathbb{C}$ be a continuous function. I Believe that $$\int_{S^{n-1}}f(x\cdot \omega)\,d\omega=\int_{S^{n-1}}f(-x\cdot \omega)\,d\omega.\qquad \qquad (1)$$ The reason ...
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37 views

Normalizing constant of an exponential family of distributions with spherical harmonics

I am interested in modeling a distribution on a sphere with a series expansion in terms of the spherical harmonics $Y_l^m (x)$ where $x \in \mathbb{S}^2$ is a point on the unit sphere. From the paper ...
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13 views

Extending fast fourier transform to spin spherical harmonic basis

I have a function that is a a Fourier transform of a product $\bar{T}(\vec{x})\partial_{\langle{i}}\partial_{j\rangle}\bar{T}^{F}(\vec{x})$ contracted with ${k}^{\langle{i}}{k}^{j\rangle}$, with the ...
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Decompose a product into angular harmonics

The function $f$ of two 3-dimensional unit vectors $\hat r_1$ and $\hat r_2$ only depends on the angle from one vector to another, so we can write $f(\hat r_1, \hat r_2)=g(\theta)$, where $\theta$ is ...
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17 views

Decompose a product into angular harmonics

The function $f$ of two 3-component unit vectors $\hat r_1$ and $\hat r_2$ only depends on the angle from one vector to the other, so we can write $f(\hat r_1, \hat r_2)=g(\theta)$, where $\theta$ is ...
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30 views

Spherical functions - symbol in solution [closed]

What does the symbol on the right-hand side of eq. (4.9) mean, please?
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Find Fourier Transform of the even function

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\right)dy\frac{dt}{t^{...
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52 views

Degree of spherical polynomial

How is it defined? For example in dimension 2, $P(x,y) = x^2 + y^2$ is a bivariate polynomial of algebraic degree 2, but when $(x,y)$ is a point of the unit circle we have $P(x,y) \equiv 1$ which is ...
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120 views

Spherical Harmonics expansion

In the contex of $L^2$ space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=...
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36 views

Angular integration of $\mathbf{\hat{r}}$ with Spherical Harmonics

I am having to an integration, $$\int_0^{2\pi}\int_0^\pi\mathbf{\hat{r}}Y^*_{\ell m}(\theta,\phi)Y_{\ell m}(\theta,\phi) \sin{\theta}d\theta d\phi.$$ Now, we know that the unit vector $\mathbf{\hat{r}}...
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41 views

Spherical average of a product of two vector lengths

Let $\vec r_1$, $\vec r_2$, and $\vec r$ be vectors in the Cartesian space. The spherical average $\int |\vec r_1 -\vec r| \, d\Omega$ (where $\Omega$ refers to $\vec r$) readily evaluates to $\frac{...
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41 views

Legendre Expansions for Derivatives of Delta Function

Expansions of the delta functions, of following type which are called completeness relations is very useful in many problems in physics. $\delta(x-y) = \sum_{n=0}^\infty P_n(x)P_n(y) \frac{2n+1}{2} $ ...
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1answer
101 views

Proving a harmonic polynomial in $x$ and $y$ is a linear combination of $\Re(x+\mathrm iy)^n$ and $\Im(x+\mathrm iy)^n$

Prove that a harmonic polynomial in $x$ and $y$ is a linear combination of $\Re(x+\mathrm iy)^n$ and $\Im(x+\mathrm iy)^n$. My train of thought is as follows: Prove that a harmonic polynomial is a ...
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I am creating an interdimensional spectral analysis tool and for that I need fast high L spherical harmonics approximation

I am creating an interdimensional spectral analysis tool and for that I need a fast high $L$ spherical harmonics approximation. I don't have a feeling for the high L evolution of spherical harmonics ...
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1answer
109 views

Volume Integral of over a Tetrahedron in Cartesian Coordinates

Evaluating a general volume integral exactly, in Cartesian coordinates, on a grid of cubic voxels is relatively easy: $$ G_i = \int_{X_{min}}^{X_{max}}\int_{Y_{min}}^{Y_{max}}\int_{Z_{min}}^{Z_{max}} ...
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1answer
223 views

Spherical harmonics and irreducible representations of $SO(2)$ and $SO(3)$

In the Wikipedia article it is mentioned (without source) that the spherical harmonics of degree $\ell$ on the $n$-sphere are an irreducible (wether real or complex is not mentioned) representations ...
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What are the matrix coefficients of the $SO(3)$ irrep of dim $k = 2m + 1$?

Peter-Weyl tells us that the matrix coefficients of the irreps of $G$ are dense in $L^2(G)$. In the specific case of $G=SO(3)$, what are these coefficients? Observation: Looking at the fundamental ...
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Hyperspherical harmonics

Consider the hyperspherical harmonics $Y_{jlm}(\phi,\theta,\varphi)$ where $\{\phi,\theta,\varphi\}$ are the angular cooridinates on the unit $S^3$. What is the following sum $$\sum_j Y^*_{jlm}(\phi=\...
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Is there a real version of the non-commutative Fourier transform?

In David Applebaum's "Probability on Compact Lie Groups", ch.2 page 36, we have the following definition of the non-commutative Fourier transform where $G$ is e.g. a compact Lie group, $f \in L^1(G; \...
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Reflection of spherical harmonics on the xy-plane

Is there a way to rewrite the reflection of the spherical harmonics on the xy-plane as a linear combination of the not reflected counterpart? Specifically, I want to find a way (if possible) to ...
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“eigenspaces of the Beltrami operator are one-dimensional.” what does this mean?

While going through the book on ellipsoidal harmonics, I came across this argument, The spectral form of the ellipsoidal Beltrami operator: $$Be(ρ)S_m^ n (μ, ν) = [(h_2^3 + h_2^2)p_m^n− n(n + 1)ρ^2]...
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Spherical Harmonics Expansion of Analytic Function derivation

Someone has led me to understand that the following spherical harmonic expansion of an analytic function is completely general, however I am having trouble seeing how one would derive it. As far as I ...
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22 views

Largest observable/calculable power spectrum multipole ($C_l$) of spherical harmonics

To understand the power spectrum in spherical harmonics, I tried to create a simple exercise using Random Gaussian variables in python. I assigned a random variable to every combination of theta and ...
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How to prove that Spherical Harmonics must have integer order $m$ and degree $n$?

Spherical harmonics : \begin{equation} Y_{m}^{n}(\theta,\phi) = N_{}\mathcal{P}_{m}^{n}(\cos \phi) e^{in\theta}, \end{equation} are the solution of the two angular equations (Legendre associated eq ...
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Prove columns of Jacobian Matrix of $T(r,\varphi,\theta) := (r\ \cos \varphi \cos \theta ,\ r\ \sin \varphi \cos \theta ,r\ \sin \theta)$ orthogonal

Let $f:\mathbb{R^3} \to \mathbb{R}$ be a differentiable function. For $r > 0, \varphi \in [0,2\pi]$ and $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ we look at the functions $$T(r,\varphi,\theta) ...
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34 views

Conventions used with Legendre Polynomials in spherical harmonics.

I have used Ambisonics audio for 10 years. I have a grasp of the maths on a trigonometric level and have spent the last six months studying Stroud's Advanced Engineering Mathematics and am learning ...
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1answer
170 views

Half sphere weighted average of spherical harmonics

One can find a few formulas for integrating combinations of spherical harmonics $Y_l^m(\theta,\phi)$ over the whole sphere. But I want to calculate the upper half sphere average of $\hat n \cdot \hat ...

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