  The first part of this paper is devoted to an analysis of moment problems in
R^n with supports contained in a closed set defined by finitely many polynomial
inequalities. The second part of the paper uses the representation results of
positive functionals on certain spaces of rational functions developed in the
first part, for decomposing a polynomial which is positive on such a
semi-algebraic set into a canonical sum of squares of rational functions times
explicit multipliers.
