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An algorithm is introduced for the rapid evaluation at appropriately chosen nodes on the two-dimensional sphere S² in R³ of functions specified by their spherical harmonic expansions (known as the inverse spherical harmonic transform), and for the evaluation of the coefficients in spherical harmonic expansions of functions specified by their values at appropriately chosen points on S² (known as the forward spherical harmonic transform). The procedure is numerically stable and requires an amount of CPU time proportional to N² (N) (1/), where N² is the number of nodes in the discretization of S², and is the precision of computations. The performance of the algorithm is illustrated via several numerical examples.
Rokhlin et al. (Sun,) studied this question.