  In this paper, we present a generalization of the classical error bounds of
Greengard-Rokhlin for the Fast Multipole Method (FMM) for Laplace potentials in
three dimensions, extended to the case of local expansion (instead of point)
targets. We also present a complementary, less sharp error bound proven via
approximation theory whose applicability is not restricted to Laplace
potentials. Our study is motivated by the GIGAQBX FMM, an algorithm for the
fast, high-order accurate evaluation of layer potentials near and on the source
layer. GIGAQBX is based on the FMM, but unlike a conventional FMM, which is
designed to evaluate potentials at point-shaped targets, GIGAQBX evaluates
local expansions of potentials at ball-shaped targets. Although the accuracy
(or the acceleration error, i.e., error due to the approximation of the
potential by the fast algorithm) of the conventional FMM is well understood,
the acceleration error of FMM-based algorithms applied to the evaluation of
local expansions has not been as well studied. The main contribution of this
paper is a proof of a set of hypotheses first demonstrated numerically in the
paper "A Fast Algorithm for Quadrature by Expansion in Three Dimensions," which
pertain to the accuracy of FMM approximation of local expansions of Laplace
potentials in three dimensions. These hypotheses are also essential to the
three-dimensional error bound for GIGAQBX, which was previously stated
conditionally on their truth and can now be stated unconditionally.
