  The paper analyses a spectral approach to reconstructing %the image of a
scalar field on the sphere, given only information about a masked version of
the field together with precise information about the (smooth) mask. The theory
is developed for a general mask, and later specialized to the case of an
axially symmetric mask. Numerical experiments are given for the case of an
axial mask motivated by the cosmic microwave background, assuming that the
underlying field is a realization of a Gaussian random field with an artificial
angular power spectrum of moderate degree ($\ell \le 100$). The recovery is
highly satisfactory in the absence of noise and even in the presence of
moderate noise.
