Abstract
In order to use the method of (least squares) collocation for the computation of an approximation to the anomalous potential of the Earth (T) it is necessary to specify a reproducing kernel Hilbert space the dual of which contain the (linear) functionals associated with the observations.
The specification includes the prescription of an inner product or an equivalent norm. It is demonstrated, that this is equivalent to the prescription of a specific reproducing kernel when an orthogonal (but not necessarily orthonormal), countable basis is known.
When T is an element of the Hilbert space, it is proved, that absolute error bounds may be computed, provided the norm of T is known. Also the convergence of a sequence of approximations obtained using observations increasing in a regular fashion is secured in this case as proved by Moritz.
In Geodetic practice the empirical covariance function of the anomalous potential has been used as a reproducing kernel and has in connection with the set of solid spherical harmonics specified a norm. It is proved, that a Hilbert space (of infinite dimension) equipped with this norm does not contain the anomalous potential.
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Tscherning, C.C. A note on the choice of norm when using collocation for the computation of approximations to the anomalous potential. Bull. Geodesique 51, 137–147 (1977). https://doi.org/10.1007/BF02522283
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DOI: https://doi.org/10.1007/BF02522283