  The evaluation of long-range potentials in periodic, many-body systems arises
as a necessary step in the numerical modeling of a multitude of interesting
physical problems. Direct evaluation of these potentials requires O(N^2)
operations and O(N^2) storage, where N is the number of interacting bodies. In
this work, we present a method, which requires O(N) operations and O(N)
storage, for the evaluation of periodic Helmholtz, Coulomb, and Yukawa
potentials with periodicity in 1-, 2-, and 3-dimensions, using the method of
Accelerated Cartesian Expansions (ACE). We present all aspects necessary to
effect this acceleration within the framework of ACE including the necessary
translation operators, and appropriately modifying the hierarchical
computational algorithm. We also present several results that validate the
efficacy of this method with respect to both error convergence and cost
scaling, and derive error bounds for one exemplary potential.
