Moment problems related to intrinsic characterizations of the moment functionals
Dragu Atanasiu
TL;DR
The paper develops intrinsic, data-light criteria for when a linear functional on a real unital algebra A is a moment functional on compact sets of characters, even when the functional is not PSD but positive on an archimedean cone. It introduces bounds C_a and associated sets K, along with Q_L, to characterize moment functionals and proves a Positivstellensatz for archimedean cones not arising from quadratic modules or semirings. The authors present alternative intrinsic characterizations, establish equivalences with existing MSTP-type results, and derive diverse applications including ball and simplex moment problems, Schmüdgen-type theorems, and semigroup-with-involution extensions. Collectively, the work broadens the toolkit for moment problems by tying positivity on structured cones to concrete representing measures under compact supports, with implications for both theory and applications in real algebraic geometry. The results unify and extend classical moment theory to more general algebraic settings and provide practical certificates for the existence and support of representing measures.
Abstract
In this paper, we consider linear functionals defined on an unital commutative real algebra A and establish characterizations for moment functionals on compact sets of characters that depend only on the given functional. For example, we obtain a characterization of a moment functional on a product of symmetric intervals, in which we do not assume that the functional is positive semidefinite but positive on a semiring of A, and a characterization of a moment functional that is a solution to the moment problem on a product of arbitrary intervals. We also prove a Positivstellensatz for an archimedean cone, which is neither a quadratic module nor a semiring.