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

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Summary

Let S be the (regular) boundary-surface of an exterior regionE e 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 containingE e ∪ 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.

Consequences of this result are versions of Runge and Keldysh-Lavrentiev theorems adapted to the chosen system {φn} and the mathematical justification of the use of trial functions in numerical (especially: collocational) procedures.

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  1. Institut für Reine und Angewandte Mathematik, Rheinisch-Westfälische Technische Hochschule Aachen, Templergraben 55, 51, Aachen, Fed. Rep. of Germany

    Willi Freeden

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  1. Willi Freeden
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Freeden, W. On the approximation of external gravitational potential with closed systems of (trial) functions. Bull. Geodesique 54, 1–20 (1980). https://doi.org/10.1007/BF02521092

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