A Note on the Choice of Norm When Using Collocation for the Computation of Approximations to the Anomalous Potential

In order to use the method of(least squares) collocation for the computation o f an approximation to the anomalous potential o f the Earth (T) it is necessary to specify a reproducing kernel Hilbert space the dual o.f which contain the (linear) functionals associated with the observation~ The specification includes the prescription o f an inner product or an equivalent norrrL It is demonstrated, that this is equivalent to the prescription o f a specij~w reproducing kernel when an orthogonal (but not necessarily orthonormal), countable b~ffs is known When T is an element o f the Hilbert space, i t is proved, that absolute error bounds may be computed, provided the norm o f T is knowtt Also the convergence o f a sequence o f 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 o f the anomalous pocential has been used as a reproducing kernel and has in connection with the set o f solid spherical harmonics specified a norrrL I t is proved, that a Hilbert space (of infinite dimension) equipped with this norm does not contain the anomalous potential.