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.