Flat Extensions of Positive Moment Matrices: Relations in Analytic or Conjugate Terms

Abstract

Given a doubly indexed finite sequence of complex numbers γ ≡ γ (2n): γ00, γ01, γ10, …, γ0,2n, …, γ2n,0, with γ00 > 0 and the truncated complex moment problem entails finding a positive Borel measure µ supported in the complex plane C such that γ is called a truncated moment sequence (of order 2n) and µ is called a representing measure for γ. The truncated complex moment problem is closely related to several other moment problems: the full moment problem prescribes moments of all orders, i.e., γ = (γij)i,j≥0, γ00 >0, the K-moment problem (truncated or full) prescribes a closed set K ⊆ C which is to contain the support of the representing measure ([Atz], [BM], [Cas], [CP], [P3], [Sch2], [StSz], [Sza]); and the multidimensional moment problem extends each of these problems to measures supported in C k ([Ber], [BCJ], [Cas], [Fug], [Havl], [Hav2], [McG], [P1], [P2], [P4]); moreover, the k-dimensional complex moment problem is equivalent to the 2k-dimensional real moment problem [CF4, Section 6]. All of these problems generalize classical power moment problems on the real line, whose study was initiated by Stieltjes, Riesz, Hamburger, and Hausdorff (cf. [AK], [Akh], [Hau], [KrN], [Lan], [Sar], [ShT]). Recently, J. Stochel [Sto] proved that a solution to the multidimensional truncated K-moment problem actually implies a solution to the corresponding full moment problem. For k = 1, we may informally paraphrase Stochel’s result as follows: If K ⊆ C is closed, if γ = (γij)i,j>0 is a full moment sequence, and if for each n ≥ 1 there exists a representing measure µ n for γ ij 0≤i+j≤2n such that supp µ n ⊆ K, then there exists a subsequence of µ n that converges (in an appropriate weak topology) to a representing measure µ for γ with supp µ ⊆ K.