Dimensional stability in truncated moment problems
The truncated multi-dimensional moment problem consists in the following: given a multi-sequence \((\gamma_\alpha)_{|\alpha|\leq 2m}\), one looks for a positive measure \(\mu\) on \({\mathbb R}^n\) such that \(\int t^\alpha d\mu=\gamma_\alpha\) for all monomials \(t^\alpha\) with \(\alpha\) a multiindex of length \(|\alpha|\leq 2m\). Clearly, the nonnegativity of the Hankel matrix \(M_\gamma=(\gamma_{\alpha+\beta})_{|\alpha|,|\beta|\leq m}\) is necessary for the solvability of the problem. Sufficient conditions for the existence of a solution of the problem with a finite support were found in [\textit{R. Curto} and \textit{L. Fialkow}, Mem. Am. Math. Soc. 568 (1996; Zbl 0876.30033)] in terms of \textit{flatness}, i.e., the existence of a rank preserving nonnegative extension \(M_{\gamma'}\) of the matrix \(M_\gamma\). In the present paper, some results of that paper concerning the existence of \(d\)-atomic solutions of the truncated moment problem are formulated and proved in terms of the dimensional stability of the so-called Riesz functional \(\Lambda\) defined on the set \({\mathcal P}\) of polynomials by \(\Lambda(t^\alpha)=\gamma_\alpha\). Let \({\mathcal P}_k\) be the set of polynomials of formal degree \(k\), \({\mathcal I}_k:=\{p\in{\mathcal P}_k: \Lambda(|p|^2)=0 \}\), \({\mathcal H}_k={\mathcal P}_k/{\mathcal I}_k\), and let \(J_k\) be a map from \({\mathcal H}_k\) to \({\mathcal H}_{k+1}\), given by \(J_k:p+{\mathcal I}_k\mapsto p+{\mathcal I}_{k+1}\), \(p\in{\mathcal P}_k\). The functional \(\Lambda\) is called \textit{dimensionally stable} if \(J_k{\mathcal H}_k={\mathcal H}_{k+1}\) for some \(k\). It is shown that a positive functional \(\Lambda\) has a \(d\)-atomic representing measure with a finite \(d\) if and only if \(\Lambda\) is dimensionally stable. The existence of a representing measure with a prescribed finite support is also studied.
scientific article; zbMATH DE number 6005685
scientific article