Satellites mapping the spatial variations of the gravitational or magnetic fields of the Earth or other planets ideally fly on polar orbits, uniformly covering the entire globe. Thus, potential fields on the sphere are usually expressed in spherical harmonics, basis functions with global support. For various reasons, however, inclined orbits are favorable. These leave a "polar gap" : an antipodal pair of axisymmetric polar caps without any data coverage, typically smaller than 10° in diameter for terrestrial gravitational problems, but 20° or more in some planetary magnetic configurations. The estimation of spherical harmonic field coefficients from an incompletely sampled sphere is prone to error, since the spherical harmonics are not orthogonal over the partial domain of the cut sphere. Although approaches based on wavelets have gained in popularity in the last decade, we present a method for localized spherical analysis that is firmly rooted in spherical harmonics. We construct a basis of bandlimited spherical functions that have the majority of their energy concentrated in a subdomain of the unit sphere by solving Slepian's (1960) concentration problem in spherical geometry, and use them for the geodetic problem at hand. Most of this work has been published by us elsewhere. Here, we highlight the connection of the "spherical Slepian basis" to wavelets by showing their asymptotic self-similarity, and focus on the computational considerations of calculating concentrated basis functions on irregularly shaped domains.