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Speaker:
Dr. Maria Infusino (University of Cagliari)

This event is part of an event series „Konstanz Women in Mathematics“.

In this talk we investigate under which conditions a linear functional L defined on a linear subspace B of a unital commutative real algebra A admits an integral representation with respect to a nonnegative Radon measure supported on a prescribed closed subset K of the space of homomorphisms of A endowed with the weak topology. This is a generalization of the classical truncated moment problem and it has the advantage of encompassing also infinite dimensional instances, e.g. when A is not finitely generated and B is finite dimensional or when A is finitely generated algebra and B infinite dimensional subspace of A.  

We first consider the case when A is equipped with a submultiplicative seminorm and provide a criterion for the existence of an integral representation for L. Then we build on this result to prove a Riesz-Haviland type theorem, which also holds when A is not necessarily equipped with a topology. This theorem allows us to extend some well-known results for the truncated moment problem for polynomials in finitely many variables to situations when the monomial diagram associated to B contains infinitely many monomials in one of the variables, e.g. for rectangular or sparse truncated moment problems. 

We also apply our result to the moment problem for point processes, which is particularly interesting for its applications in statical mechanics though very little is known in literature about its solvability. Last but not least, we closely analyze the relation between the full and the truncated moment problem in our general setting, obtaining a result in the spirit of Stochel’s theorem, which easily applies to full moment problems for localized algebras. 

This is a joint work with Raúl Curto, Mehdi Ghasemi, and Salma Kuhlmann.