Positive linear functionals without representing measures

Abstract

For k even, let Pk denote the vector space of polynomials in 2 real variables of degree at most k. A linear functional L : Pk −→ R is positive if p ∈ Pk, p|R 2 0 =⇒ L(p) 0. Hilbert's theorem on sums of squares (cf. (15)) implies that L : P4 −→ R is positive if and only if the moment matrix associated to L is positive semidefinite. In this note, using k = 6, we exhibit the first family of positive linear functionals L : Pk → R whose positivity cannot be derived from the positive semidefiniteness of the associated moment matrices, and which do not correspond to integration with respect to positive measures.