  Problems in $\mathbb{R}^3$ are addressed where the scalar potential of an
associated vector field satisfies Laplace's equation in some unbounded external
region and is to be approximated by unknown (point) sources contained in the
complimentary subregion. Two specific field geometries are considered:
$\mathbb{R}^3$ half-space and the exterior of an $\mathbb{R}^3$ sphere, which
are the two standard settings for geophysical and geoexploration gravitational
problems. For these geometries it is shown that a new type of kernel space
exists, which is labeled a Dirichlet-integral dual-access collocation-kernel
space (DIDACKS) and that is well suited for many applications. The DIDACKS
examples studied are related to reproducing kernel Hilbert spaces and they have
a replicating kernel (as opposed to a reproducing kernel) that has the
ubiquitous form of the inverse of the distance between a field point and a
corresponding source point. Underpinning this approach are three basic
mathematical relationships of general interest. Two of these
relationships--corresponding to the two geometries--yield exact closed-form
inner products and thus exact linear equation sets for the corresponding point
source strengths of various types (i.e., point mass, point dipole and/or point
quadrupole sets) at specified source locations. The given field is
reconstructed not only in a point collocation sense, but also in a (weighted)
field-energy error-minimization sense.
