The realizability problem as a special case of the infinite-dimensional truncated moment problem

The realizability problem is a well-known problem in the analysis of complex systems, which can be modeled as an infinite-dimensional moment problem. More precisely, as a truncated K-moment problem where K is the space of all possible configurations of the components of the considered system. The power of this reformulation has been already exploited by Kuna, Lebowitz, and Speer [Ann. Appl. Probab. 21 (2011), pp. 1253-1281], where necessary and sufficient conditions of Haviland type have been obtained for several instances of the realizability problem. In this article we exploit this same reformulation to apply to the realizability problem the recent advances obtained by Curto, Ghasemi, Infusino, and Kuhlmann [J. Operator Theory 90 (2023), pp. 223-261] for the truncated moment problem for linear functionals on general unital commutative algebras. This provides alternative proofs and sometimes extensions of several results of Kuna, Lebowitz, and Speer [Ann. Appl. Probab. 21 (2011), pp. 1253-1281], allowing to finally embed them in the above-mentioned unified framework for the infinite-dimensional truncated moment problem.