Natively Periodic Fast Multipole Method: Approximating the Optimal Green’s Function

The Fast Multipole Method (FMM) obeys periodic boundary conditions "natively" if it uses a Green function for computing the multipole expansion in interaction zone of each FMM oct-tree node. One can define "optimal" such method that results numerical solution converges to equivalent Particle-Mesh limit sufficiently high order multipoles. A discrete functional equation optimal be derived, but is not practically useful as methods its are known. Instead, this paper presents an approximation accurate better than 1e-3 LMAX norm and 1e-4 L2 counts. Such approximately offers practical way implementing with "natively", without need compute lattice sums or rely on hybrid FMM-PM approaches.