A fast and stable method for rotating spherical harmonic expansions
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.
scientific article; zbMATH DE number 5596637
scientific article