The recovery of the harmonic coefficients of the anomalous potential from a geodetic quantity sampled over a regular grid is affected by the non-exact discrete orthogonality of spherical harmonics; larger errors occur for block-average quantities owing to the non-simple behaviour of the block-average operator when applied to spherical harmonics. Fourier coefficients, on the contrary, can be recovered by exact formulas both from point data and block-averaged data; furthermore, Fourier analysis enables to give a rigorous description of aliasing from high-degree harmonic components for the sphere. The harmonic coefficients of a truncated model can then be obtained by solving linear systems. Numerical simulations confirm the advantages of the method.