A fast and stable method for rotating spherical harmonic expansions
type: Item
instance of: scholarly article
title: A fast and stable method for rotating spherical harmonic expansions
zbMATH Open document ID: 1184.33007
published in: Journal of Computational Physics
publication date: 2009-08-19T00:00:00Z
review text: A method for rotating a spherical harmonic expansion is considered. Let \[ u(\theta,\varphi)=\sum_{n=0}^\infty\sum_{m=-n}^n M_n^mY_n^m(\theta,\varphi) \] be a harmonic function on the unit sphere, where \(Y_n^m(\theta,\varphi)\) is the spherical harmonic of degree \(n\) and order \(m\). It is known [\textit{L. C. Biedenharn} and \textit{J. D. Louck}, Angular momentum in quantum physics. Theory and application. With a foreword by Peter A. Carruthers. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1981; Zbl 0474.00023)] that if \((\theta,\varphi)\) denotes the coordinates of a point \(P\) in the usual coordinate system \(E=\{e_1,e_2,e_3\}\) and \((\theta',\varphi')\) denotes the coordinates of \(P\) in the rotated orthogonal coordinate system with the same origin \(F=\{f_1,f_2,f_3\}\), then the harmonic function \(u(P)\) can be expressed as \(u(\theta',\varphi')=\sum_{n=0}^\infty\sum_{m'=-n}^n M_n^{m'}Y_n^{m'}(\theta',\varphi')\), where \(M_n^{m'}=\sum_{m=-n}^n D_n^{m',m}M_n^{m}\) and \(D_n^{m',m}\) are explicitly given. A scheme for rotating a spherical harmonic expansion of arbitrary degree \(p\) based on ``pseudospectral'' projection method rather than the explicit construction of the rotation matrix \((D_n^{m',m})\) is presented.
reviewed by: Valery V. Karachik
Mathematics Subject Classification ID: 33C55
Mathematics Subject Classification ID: 35C10
zbMATH DE Number: 5596637
zbMATH Keywords: rotation matrix
zbMATH Keywords: spherical harmonics
zbMATH Keywords: fast multipole method
author: Zydrunas Gimbutas
author: Leslie Greengard
MaRDI profile type: Publication
full work available at URL: j.jcp.2009.05.014
OpenAlex ID: W2122146619
cites work: Q3929699
cites work: Q4838168
cites work: A wideband fast multipole method for the Helmholtz equation in three dimensions
cites work: Q3246876
cites work: Q4356574
cites work: Recursions for the Computation of Multipole Translation and Rotation Coefficients for the 3-D Helmholtz Equation
cites work: Q5722964
cites work: Q5476874
cites work: Rotation matrices for real spherical harmonics: general rotations of atomic orbitals in space-fixed axes
cites work: Q3243822
DOI: 10.1016/J.JCP.2009.05.014
Recommended article: Efficient and accurate rotation of finite spherical harmonics expansions
Recommended article: A representation for spherical harmonics of a rotated argument
Recommended article: Rotation of real spherical harmonics
Recommended article: A Simple Proof of Fryant’s Theorem
Recommended article: Transformation of complex spherical harmonics under rotations
Recommended article: Q3329672
Recommended article: Recursive Computation of Spherical Harmonic Rotation Coefficients of Large Degree
Recommended article: Spherical harmonics in a non‐polar co‐ordinate system and application to Fourier series in 2‐sphere
Recommended article: Constructing the spherical harmonics
Recommended article: Global spherical harmonic computation by two-dimensional Fourier methods
description: scientific article; zbMATH DE number 5596637