  We consider the global minimization of a polynomial on a compact set B. We
show that each step of the Moment-SOS hierarchy has a nice and simple
interpretation that complements the usual one. Namely, it computes coefficients
of a polynomial in an orthonormal basis of L 2 (B, $\mu$) where $\mu$ is an
arbitrary reference measure whose support is exactly B. The resulting
polynomial is a certain density (with respect to $\mu$) of some signed measure
on B. When some relaxation is exact (which generically takes place) the
coefficients of the optimal polynomial density are values of orthonormal
polynomials at the global minimizer and the optimal (signed) density is simply
related to the Christoffel-Darboux (CD) kernel and the Christoffel function
associated with $\mu$. In contrast to the hierarchy of upper bounds which
computes positive densities, the global optimum can be achieved exactly as
integration against a polynomial (signed) density because the CD-kernel is a
reproducing kernel, and so can mimic a Dirac measure (as long as finitely many
moments are concerned).
