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A new acceleration technique with exponential convergence rate to evaluate periodic Green functions

We propose a new approach to compute the dynamic potential (Green function) from two- and three-dimensional periodic arrays of point sources in a three-dimensional space. The method has at least exponential convergence rate (in some cases it has Gaussian convergence rate). The convergence rate is independent of the position of the observation point in the unit cell. The proposed approach can also be used to calculate the derivatives of the Green function very efficiently.