  In the field of orthogonal polynomials theory, the classical Markov theorem
shows that for determinate moment problems the spectral measure is under
control of the polynomials asymptotics. The situation is completely different
for indeterminate moment problems, in which case the interesting spectral
measures are to be constructed using Nevanlinna theory. Nevertheless it is
interesting to observe that some spectral measures can still be obtained from
weaker forms of Markov theorem. The exposition will be illustrated by
orthogonal polynomials related to elliptic functions: in the determinate case
by examples due to Stieltjes and some of their generalizations and in the
indeterminate case by more recent examples.
