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 constitute an orthonormal. basis of a subset of R (or R-2) concentrating the maximum information in the subspace of square integrable functions with a band-limited spectrum. The same problem can be stated and solved on the sphere. The relation between the new basis and the ordinary spherical harmonic basis can be explicitly written and numerically studied. The new base functions are orthogonal on both the subspace and the whole sphere. Numerical tests show the applicability of the Slepian approach with regard to solvability and stability in the case of polar data gaps, even in the presence of aliasing. This tool turns out to be a natural solution to the polar gap problem in satellite geodesy. It enables capture of the maximum amount of information from non-polar gravity field missions.