On the approximation of external gravitational potential with closed systems of (trial) functions

TLDR: In this article, it was shown that the restrictions of the functions on a regular boundary surface of an exterior region in Euclidean space ℜ3 (for instance, sphere, ellipsoid, geoid, earth's surface) form a closed system in the spaceC (S), i.e. any functionf, defined and continuous on S, can be approximated uniformly by a linear combination of functions ϕn.

Abstract: Let S be the (regular) boundary-surface of an exterior regionEe in Euclidean space ℜ3 (for instance: sphere, ellipsoid, geoid, earth's surface). Denote by {φn} a countable, linearly independent system of trial functions (e.g., solid spherical harmonics or certain singularity functions) which are harmonic in some domain containingEe ∪ S. It is the purpose of this paper to show that the restrictions {ϕn} of the functions {φn} onS form a closed system in the spaceC (S), i.e. any functionf, defined and continuous onS, can be approximated uniformly by a linear combination of the functions ϕn.