This paper introduces a fast algorithm for obtaining a uniform resolution representation of a function known at a latitude-Swarztrauber [12] has reviewed other transformation algolongitude grid on the surface of a sphere, equivalent to a triangular, rithms.
isotropic truncation of the spherical harmonic coefficients for the In this paper we introduce a technique for avoiding the function. The proposed spectral truncation method, which is based surface harmonics transformation, while retaining the benon the fast multipole method and the fast Fourier transform, projects the function to a space with uniform resolution while avoiding sur-efits of using surface harmonics representations. The techface harmonic transformations. The method requires O(N 2 log N) nique is based on a fast algorithm for the orthogonal prooperations for O(N 2 ) grid points, as opposed to O(N 3 ) operations jection of a function defined on the sphere and known at for the standard spectral transform method, providing a reduceda latitude-longitude grid onto the space spanned by a complexity spectral method obviating the pole problem in the integration of time-dependent partial differential equations on the truncated surface harmonic expansion. Appropriate choice sphere. The filter's performance is demonstrated with numerical of the truncation gives representations with uniform resoexamples. ᮊ 1997 Academic Press lution. The algorithm uses the Christoffel-Darboux formula [14] for the summation of products of orthogonal polynomials, in combination with the fast multipole method [5] and the fast Fourier transform. Semi-implicit
2. THE SPHERICAL FILTER
[3] and, more recently, Healy, Moore, and Rockmore have proposed a fast surface harmonics transformation algorithm of asymptotic complexity O(N 2 log 2 N ), but This section consists of three subsections: in the first we some questions remain regarding its efficiency and stabilmathematically define the spherical filter; in the second we ity. Alpert and Rokhlin [1] have described a fast algorithm summarize the standard spectral transform algorithm; and in the third we describe the novel spectral truncation method. † Deceased.