THE PROBLEM OF MOMENTS ON COMPACT SEMI-AL GEBRAIC SETS

Let K = {x is an element of R(n); p(1)(x) greater than or equal to 0, ..., p(N)(x) greater than or equal to 0} be a compact semi-algebraic s ubset of Rn, where p(1), ..., p(N) are polynomials normalized by paral lel to p(j) parallel to(infinity,K) less than or equal to 1 (1 less th an or equal to j less than or equal to N, N greater than or equal to n + 1), and such that p(1), ... pn are of degree one and linearly indep endent. Then the problem of moments a(alpha) = integral(K) x(alpha)d m u(x), alpha is an element of Nn, has as a solution a positive Borel me asure mu on K if and only if the associated functional L is an element of R[x]', L(x(alpha)) = a(alpha)is an element of Nn), is nonnegative on the set of polynomials of the form p(1)(m1)... p(N)(mN) (1-pN)(kN), where m(1), ..., m(N), k(1), ... k(N) are arbitrary nonnegative integ ers.