Abstract
We develop a multiresolution representation of a class of integral operators satisfying
boundary conditions on simple domains in order to construct fast algorithms for their application.
We also elucidate some delicate theoretical issues related to the construction of periodic Green’s
functions for Poisson’s equation.
By applying the method of images to the non-standard form of the free space operator, we
obtain lattice sums that converge absolutely on all scales, except possibly on the coarsest scale. On
the coarsest scale the lattice sums may be only conditionally convergent and, thus, allow for some
freedom in their definition. We use the limit of square partial sums as a definition of the limit and
obtain a systematic, simple approach to the construction (in any dimension) of periodized operators
with sparse non-standard forms.
We illustrate the results on several examples in dimensions one and three: the Hilbert transform,
the projector on divergence free functions, the non-oscillatory Helmholtz Green’s function and the
Poisson operator. Remarkably, the limit of square partial sums yields a periodic Poisson Green’s
function which is not a convolution.
Using a short sum of decaying Gaussians to approximate periodic Green’s functions, we arrive at
fast algorithms for their application. We further show that the results obtained for operators with
periodic boundary conditions extend to operators with Dirichlet, Neumann, or mixed boundary
conditions.