About the Importance of the Runge–Walsh Concept for Gravitational Field Determination

Abstract.

On the one hand, the Runge–Walsh theorem plays a particular role in physical geodesy, because it allows to guarantee a uniform approximation of the Earth’s gravitational potential within arbitrary accuracy by a harmonic function showing a larger analyticity domain. On the other hand, there are some less transparent manifestations of the Runge–Walsh context in the geodetic literature that must be clarified in more detail. Indeed, some authors make the attempt to apply the Runge–Walsh idea to the gravity potential of a rotating Earth instead of the gravitational potential in non-rotating status. Others doubt about the convergence of series expansions approximating the Earth’s gravitational potential inside the whole outer space of the actual Earth.

The goal of this contribution is to provide the conceptual setup of the Runge–Walsh theorem such that geodetic expectation as well as mathematical justification become transparent and coincident. Even more, the Runge–Walsh concept in form of generalized Fourier expansions corresponding to certain harmonic trial functions (e.g., mono- and/or multi-poles) will be extended to the topology of Sobolev-like reproducing kernel Hilbert spaces thereby avoiding any need of (numerical) integration in the occurring spline solution process.