Approximating Positive Polynomials Using Sums of Squares

XS f dμ. It is natural to ask if the same is true for any linear functional L : R[X ] → R which is non-negative on MS. This is the Moment Problem for the quadratic module MS. The most interesting case seems to be when S is finite. A sufficient condition for it to be true is that each f ∈ T̃S can be approximated by elements of MS in the sense that there exists an element q ∈ R[X ] such that, for all rational ǫ > 0,