Computation of spherical harmonic expansion coefficients via FFT's (Q1057048)

The formulas for the coefficients of an expansion of a function f in terms of spherical harmonics \(Y_{\ell m}\) are obtained by assuming that both f and \(Y_{lm}\) are represented by trigonometric polynomials, and by evaluating the occuring integrals of sine functions. From some sufficiently large \(l=l_ N\) on, this needs fewer operations than numerical double integration with \(N^ 2\) meshpoints. Storage and error are discussed and a simple numerical example is presented. The method seems to be generally faster than the method of \textit{W. Freeden} [Computing 25, 131-146 (1980; Zbl 0419.65014)] but practically inapplicable for irregularly spaced data f over the sphere.