Infinite dimensional moment problem: open questions and applications

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.Comment: 14 pages, minor revisions according to referee's comments, updated reference

The full infinite dimensional moment problem on symmetric algebras of locally convex real spaces

This talk aims to introduce an infinite dimensional version of the classical full moment problem and explore certain instances which actually arise in several applied fields. The general theoretical question addressed is whether a linear functional $L$ on the symmetric algebra $S(V)$ of a locally convex topological real vector space $V$ can be represented as an integral w.r.t.\! a non-negative Radon measure supported on a fixed subset of the algebraic dual $V^*$ of $V$. I present a recent joint work with M. Ghasemi, S. Kuhlmann and M. Marshall where we get representations of continuous positive semidefinite linear functionals $L: S(V)\rightarrow \mathbb{R}$ as integrals w.r.t.\! uniquely determined Radon measures supported in special sorts of closed balls in the topological dual space $V\u27$ of $V$. A better characterization of the support is obtained when $L$ is positive on a $2d-$power module of $S(V)$. I compare these results with the corresponding ones for the full moment problem on locally convex nuclear spaces, pointing out the crucial roles played by the continuity and the quasi-analyticity assumptions on $L$ in determining the support of the representing measure. In particular, I focus on a joint work with T. Kuna and A. Rota where we derive an analogous result for functionals on the symmetric algebra of the space of test functions on $\mathbb{R}^d$ which are positive on quadratic modules but not necessarily continuous. This setting is indeed general enough to encompass many spaces which occur in concrete applications, e.g.\! the space of point configurations

Uniform distribution on fractals

In this paper we introduce a general algorithm to produce u.d. sequences of partitions and of points on fractals generated by an IFS consisting of similarities which have the same ratio and which satisfy the open set condition (OSC). Moreover we provide an estimate for the elementary discrepancy of van der Corput type sequences constructed on this class of fractals

On the discrepancy of some generalized Kakutani's sequences of partitions

In this paper we study a class of generalized Kakutani’s sequences of partitions of [0,1], constructed by using the technique of successive refinements. Our main focus is to derive bounds for the discrepancy of these sequences. The approach that we use is based on a tree representation of the sequence of partitions which is precisely the parsing tree generated by Khodak’s coding algorithm. With the help of this technique we derive (partly up to a logarithmic factor) optimal upper bound in the so-called rational case. The upper bounds in the irrational case that we obtain are weaker, since they heavily depend on Diophantine approximation properties of a certain irrational number. Finally, we present an application of these results to a class of fractals

Translation invariant realizability problem on the d-dimensional lattice: an explicit construction

We consider a particular instance of the truncated realizability problem on the d−dimensional lattice. Namely, given two functions ρ1(i) and ρ2(i,j) non-negative and symmetric on Zd, we ask whether they are the first two correlation functions of a translation invariant point process. We provide an explicit construction of such a realizing process for any d ≥ 2 when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds

Infinite-dimensional moment-SOS hierarchy for nonlinear partial differential equations

We formulate a class of nonlinear {evolution} partial differential equations (PDEs) as linear optimization problems on moments of positive measures supported on infinite-dimensional vector spaces. Using sums of squares (SOS) representations of polynomials in these spaces, we can prove convergence of a hierarchy of finite-dimensional semidefinite relaxations solving approximately these infinite-dimensional optimization problems. As an illustration, we report on numerical experiments for solving the heat equation subject to a nonlinear perturbation.Comment: 24 pages, 1 table, 3 figure

Projective limits techniques for the infinite dimensional moment problem

We deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra $A$ be represented as an integral with respect to a Radon measure on the character space $X(A)$ of $A$ equipped with the Borel $\sigma-$algebra generated by the weak topology $\tau_{X(A)}$? We approach this problem by constructing $X(A)$ as a projective limit of the character spaces of all finitely generated unital subalgebras of $A$ and so by considering on $X(A)$ two natural measurable structures: the associated cylinder $\sigma-$algebra and the Borel $\sigma-$algebra generated by the corresponding projective topology which coincides with the weak topology $\tau_{X(A)}$. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on $A$ with respect to a measure on the cylinder $\sigma-$algebra on $X(A)$ (resp. a Radon measure on the Borel $\sigma-$algebra on $X(A)$) provided that for any finitely generated unital subalgebra of $A$ the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on $A$. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a "partially Archimedean" quadratic module of $A$. Our results in particular apply to the case when $A$ is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing alternative proofs (and in some cases stronger versions) of some recent results for these instances of the moment problem and offering at the same time a unified setting which enables comparisons.Comment: 35 pages, 3 figure