  Smoothing is omnipresent in astronomy, because almost always measurements
performed at discrete positions in the sky need to be interpolated into a
smooth map for subsequent analysis. Still, the statistical properties of
different interpolation techniques are very poorly known. In this paper, we
consider the general problem of interpolating discrete data whose location
measurements are distributed on the sky according to a known density
distribution (with or without clustering). We derive expressions for the
expectation value and for the covariance of the smoothed map for many
interpolation techniques, and obtain a general method that can be used to
obtain these quantities for any linear smoothing. Moreover, we show that few
basic properties of smoothing procedures have important consequences on the
statistical properties of the smoothed map. Our analysis allows one to obtain
the statistical properties of an arbitrary interpolation procedure, and thus to
optimally choose the technique that is most suitable for one's needs.
