The truncated complex -moment problem

Let γ ≡ γ(2n) denote a sequence of complex numbers γ00, γ01, γ10, . . . , γ0,2n, . . . , γ2n,0 (γ00 > 0, γij = γji), and let K denote a closed subset of the complex plane C. The Truncated Complex K-Moment Problem for γ entails determining whether there exists a positive Borel measure μ on C such that γij = ∫ zizj dμ (0 ≤ i + j ≤ 2n) and suppμ ⊆ K. For K ≡ KP a semi-algebraic set determined by a collection of complex polynomials P = {pi (z, z)}mi=1, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n) (γ) and the localizing matrices Mpi . We prove that there exists a rankM (n)-atomic representing measure for γ(2n) supported in KP if and only if M (n) ≥ 0 and there is some rankpreserving extension M (n+ 1) for which Mpi (n+ ki) ≥ 0, where deg pi = 2ki or 2ki − 1 (1 ≤ i ≤ m).