We first show that a continuous function f is nonnegative on a closed set
K⊆Rn if and only if (countably many) moment matrices of some signed
measure dν=fdμ with support equal to K, are all positive semidefinite
(if K is compact ÎĽ is an arbitrary finite Borel measure with support
equal to K. In particular, we obtain a convergent explicit hierarchy of
semidefinite (outer) approximations with {\it no} lifting, of the cone of
nonnegative polynomials of degree at most d. Wen used in polynomial
optimization on certain simple closed sets \K (like e.g., the whole space
Rn, the positive orthant, a box, a simplex, or the vertices of the
hypercube), it provides a nonincreasing sequence of upper bounds which
converges to the global minimum by solving a hierarchy of semidefinite programs
with only one variable. This convergent sequence of upper bounds complements
the convergent sequence of lower bounds obtained by solving a hierarchy of
semidefinite relaxations