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