Infinite dimensional moment problem: open questions and applications

Abstract

This paper is dedicated to Murray Marshall, who posed many of the questions here addressed and was still working on them in the very last days of his life. We lost a wonderful collaborator and a dear friend. We sorely miss him.Abstract. Infinite dimensional moment problems have a long history in diverse applied areas dealing with the analysis of complex systems but progress is hindered by the lack of a general understanding of the mathematical structure behind them. Therefore, such problems have recently got great attention in real algebraic geometry also because of their deep connection to the finite dimensional case. In particular, our most recent collaboration with Murray Marshall and Mehdi Ghasemi about the infinite dimensional moment problem on symmetric algebras of locally convex spaces revealed intriguing questions and relations between real algebraic geometry, functional and harmonic analysis. Motivated by this promising interaction, the principal goal of this paper is to identify the main current challenges in the theory of the infinite dimensional moment problem and to highlight their impact in applied areas. The last advances achieved in this emerging field and briefly reviewed throughout this paper led us to several open questions which we outline here.

Second-order impulsive differential systems with mixed and several delays

“…However, the description of some real world problems requires studies on systems of differential equations with impulses, a subject very interesting from a mathematical point of view. Examples of the aforementioned phenomena are related to mechanical systems, biological systems, population dynamics, pharmacokinetics, theoretical physics, biotechnology processes, chemistry, engineering, control theory (we also stress that in the modeling of these phenomena is suitably formulated by evolutive partial differential equations and, moreover, moment problem approaches appear also as a natural instrument in control theory of neutral type systems; see [4][5][6] and [7], respectively).…”

First-order impulsive differential systems: sufficient and necessary conditions for oscillatory or asymptotic behavior

“…However, the description of some real world problems requires studies on impulsive differential systems, a subject very interesting from the mathematical point of view. Examples of the aforementioned phenomena are related to theoretical physics, pharmacokinetics, population dynamics, biotechnology processes, biological systems, mechanical systems, control theory, chemistry, engineering (we also stress that the modeling of these phenomena is suitably formulated by evolutive partial differential equations and, moreover, moment problem approaches appear also as a natural instrument in control theory of neutral type systems; see [4][5][6] and [7][8][9], respectively).…”

Projective limits techniques for the infinite dimensional moment problem

“…In [16] and, more recently, in [37], Problem 0.1 is considered for Σ being a σ-algebra smaller than the Borel σ−algebra. There has been actually lot of activity about Problem 0.1 for topological algebras and representing Radon measures (see, e.g., [4], [13], [14], [15], [17], [25], [22], [27], [34] in addition to some of the work mentioned above about S(V )). In this article we do not consider any topology on our algebras when studying Problem 0.1 and, as for the classical case, our final purpose is to find necessary and sufficient conditions for the existence of a K-representing Radon measures on X(A), i.e.…”