A generalized flat extension theorem for moment matrices (Q1039946)

The flat extension theorem of \textit{R. E. Curto} and \textit{L. A. Fialkow} [Solution of the truncated complex moment problem for flat data. Mem. Am. Math. Soc. 568, 52 p. (1996; Zbl 0876.30033)] for truncated moment matrices is generalized in this paper, in order to include moment matrices indexed by an arbitrary set of monomials. The main result of the paper is the following: Let \(x=(x_1,\dots,x_n)\) be a multi-variable, let \({\mathcal C}\subset {\mathcal M}_n\). Let \[ {\mathcal C}^+={\mathcal C}\cup\bigcup_{i=1}^n x_i {\mathcal C}=\big \{m,x_1m,\dots,x_nm\; |\; m\in {\mathcal C}\big\} \] and \(\partial {\mathcal C}={\mathcal C}^+-{\mathcal C}\) stand respectively for the closure and the border of \({\mathcal C}\). Let \((y_a)_{a\in {\mathcal C}^+\cdot {\mathcal C}^+}\) and assume that every monomial \(m\in {\mathcal C}-\{1\}\) can be written as \(m=x_{i_1}\dots x_{i_k}\) with \(x_{i_1}\), \(x_{i_1}x_{i_2}\), \(x_{i_1}x_{i_2}x_{i_3}\), \(\dots\), \(x_{i_1}\dots x_{i_k}\in {\mathcal C}\). If \(M_{\mathcal C}^+(y)\) is a flat extension of \(M_{\mathcal C}(y)\), then there exists a (unique) sequence \(\tilde{y}=(\tilde{y}_a)_{a\in {\mathcal M}_n}\) for which \(M(\tilde{y})\) is a flat extension of \(M_{\mathcal C}^+(y)\).