The analytic foundations for a new version of the fast multipole method (FMM) for the scalar Helmholtz equation in the low-frequency regime are presented. The computational cost of existing FMM implementations, is dominated by the expense of translating far-field partial wave expansions to local ones, requiring 189p4 or 189p3 operations per box, where harmonics up to order p2 have been retained. By developing a new expansion in plane waves, these translation operators can be diagonalized. The approach combines evanescent and propagating plane waves to reduce the computational cost of FMM implementation.
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