  In a number of problems in computational physics, a finite sum of kernel
functions centered at $N$ particle locations located in a box in three
dimensions must be extended by imposing periodic boundary conditions on box
boundaries. Even though the finite sum can be efficiently computed via fast
summation algorithms, such as the fast multipole method (FMM), the periodized
extension is usually treated via a different algorithm, Ewald summation,
accelerated via the fast Fourier transform (FFT). A different approach to
compute this periodized sum just using a blackbox finite fast summation
algorithm is presented in this paper. The method splits the periodized sum in
to two parts. The first, comprising the contribution of all points outside a
large sphere enclosing the box, and some of its neighbors, is approximated
inside the box by a collection of kernel functions ("sources") placed on the
surface of the sphere or using an expansion in terms of spectrally convergent
local basis functions. The second part, comprising the part inside the sphere,
and including the box and its immediate neighborhood, is treated via available
summation algorithms. The coefficients of the sources are determined by least
squares collocation of the periodicity condition of the total potential,
imposed on a circumspherical surface for the box. While the method is presented
in general, details are worked out for the case of evaluating electrostatic
potentials and forces. Results show that when used with the FMM, the periodized
sum can be computed to any specified accuracy, at an additional cost of the
order of the free-space FMM. Several technical details and efficient algorithms
for auxiliary computations are provided, as are numerical comparisons.
