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Translation of multipole and local expansions

In dealing with higher (lower) frequency or larger (smaller) domains, one needs to consider denser (sparser) set of sample points on the unit sphere to keep the accuracy of the computation of the integral in Eq. (9) independent of the level. In the M2M (L2L) translation, the size of the cell becomes larger (smaller) as we move upward (downward) in the tree structure of cells in the upward (downward) path of the FMM algorithm. Therefore the interpolation of multipole moments (the anterpolation of coefficients of the local expansion) on the unit sphere is necessary in M2M (L2L) translations. In this paper we carry out the interpolation and anterpolation using the far-field transform and its inverse transform (See Rokhlin[1] or Gyure and Stalzer[7]), as we shall see in the sequel.