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Title: From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions
Author: Rota, Aldo
Awarding Body: University of Reading
Current Institution: University of Reading
Date of Award: 2013
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We derive necessary and sufficient conditions for an infinite sequence of Radon measures to be realized by, or to be the sequence of moment functions of, a finite measure concentrated on a pre-given basic semi-algebraic subset of the space of generalized functions on Rd A set of such a kind is given by (not necessarily countable many) polynomial constraints. We get realizability conditions of semi definite type that can be more easily and efficiently verified, via semi definite programming, than the well-known Riesz-Haviland type condition. As a consequence, we characterize the support of the realizing measure in terms of its moments functions. As concrete examples of basic semi-algebraic sets of generalized functions, we present the set of all Radon measures, the set of all bounded Radon measures with Radon-Nikodym density w.r.t. the Lebesgue measure, the set of all probabilities, the set of all sub-probabilities and the set of all point configurations. These examples are considered in numerous areas of applications dealing with the description of large complex system. Our approach is based on a combination of classical results about the moment problem on nuclear spaces and of techniques developed to solve the moment problem on basic semi-algebraic sets of IRd For this reason, we provide a unified exposition of some aspects of the classical real moment problem which have inspired our main result. Particular importance is given to criteria for existence and uniqueness of the realizing measure on IRd via the multivariate Carleman condition and the operator-theoretical approach. We also give a formulation of the moment problem on general finite dimensional spaces in duality which makes clear the analogies with the infinite dimensional moment problem on nuclear spaces.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available