  Convex regularization techniques are now widespread tools for solving inverse
problems in a variety of different frameworks. In some cases, the functions to
be reconstructed are naturally viewed as realizations from random processes; an
important question is thus whether such regularization techniques preserve the
properties of the underlying probability measures. We focus here on a case
which has produced a very lively debate in the cosmological literature, namely
Gaussian and isotropic spherical random fields, and we prove that Gaussianity
and isotropy are not conserved in general under convex regularization over a
Fourier dictionary, such as the orthonormal system of spherical harmonics.
