The truncated complex $K$-moment problem

Abstract

Let γ ≡ γ (2n) denote a sequence of complex numbers γ 00 , γ 01 , γ 10 ,. .. , γ 0,2n ,. .. , γ 2n,0 (γ 00 > 0, γ ij = ¯ γ ji), 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 γ ij = ¯ z i z j dµ (0 ≤ i + j ≤ 2n) and supp µ ⊆ K. For K ≡ K P a semi-algebraic set determined by a collection of complex poly-nomials P = {p i (z, ¯ z)} m i=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 Mp i. We prove that there exists a rank M (n)-atomic representing measure for γ (2n) supported in K P if and only if M (n) ≥ 0 and there is some rank-preserving extension M (n + 1) for which Mp i (n + k i) ≥ 0, where deg p i = 2k i or 2k i − 1 (1 ≤ i ≤ m).