Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere

The Slepian problem consists of determining a sequence of functions that co nstitute an orthonormal. basis of a subset of R (or R-2) concentrating the maximum information in the subspace of square integrable functions with a b and-limited spectrum. The same problem can be stated and solved on the sphe re. The relation between the new basis and the ordinary spherical harmonic basis can be explicitly written and numerically studied. The new base funct ions are orthogonal on both the subspace and the whole sphere. Numerical te sts show the applicability of the Slepian approach with regard to solvabili ty and stability in the case of polar data gaps, even in the presence of al iasing. This tool turns out to be a natural solution to the polar gap probl em in satellite geodesy. It enables capture of the maximum amount of inform ation from non-polar gravity field missions.