IWOTA 2019

International Workshop on
Operator Theory and its Applications

July 22-26, 2019
Instituto Superior Técnico
Lisbon, Portugal


Truncated Moment Problems

Raúl E. Curto
University of Iowa

Bivariate truncated moment problems with algebraic variety in the nonnegative quadrant of $\mathbb{R}^2$

We consider truncated moment problems associated with algebraic varieties in the nonnegative quadrant of $\mathbb{R}^2$. We focus our attention on the interactions between two related problems, as follows.

  1. The Subnormal Completion Problem (SCP) for $2$-variable weighted shifts: We use tools and techniques from the theory of truncated moment problems to give a general strategy to solve SCP. We then show that when all quadratic moments are known (equivalently, when the initial segment of weights consists of five independent data points), the natural necessary conditions for the existence of a subnormal completion are also sufficient. To calculate explicitly the associated Berger measure, we compute the algebraic variety of the associated truncated moment problem; it turns out that this algebraic variety is precisely the support of the Berger measure of the subnormal completion.
  2. One-step extensions of $2$-variable weighted shifts: We provide necessary and sufficient conditions for the subnormality of such extensions, by using backward extensions, disintegration-of-measure techniques, and $k$-hyponormality techniques from the theory of $2$-variable weighted shifts. We apply our results to solve an interpolation problem for measures on $\mathbb{R}^2$.

The talk is based on joint work with S. H. Lee and J. Yoon.

Philipp di Dio
TU Berlin

Carathéodory Numbers and Reconstruction of Measures from Moments

Joint Work with Mario Kummer and Konrad Schmüdgen.

We present new lower bounds on the Caratheodory numbers and their asymthotic behavior. By Richters Theorem (1957) every truncated moment functional is a conic combination of point evaluations and the Caratheodory number is the minimal number of point evaluations required to represent a (or all) truncated moment functional(s). We show that the Caratheodory number is cursed by high dimensions, i.e., for any given degree $d$ and $a \gt 0$ there is a natural number $n$ and a truncated moment functional $L$ which needs at least $(1-a)*\binom{n+d}{n}$ point evaluations.

We introduce the concept of a derivative of a (truncated moment) functional and show how this unified theory provides easy and efficient proofs and methods to reconstruct shapes, functions, and measures from their moments.

Lawrence Fialkow
State University of New York at New Paltz

The core variety and conical representations

This is joint work with Grigoriy Blekherman (Georgia Tech. Univ.). Let $s$ be a truncated multisequence corresponding to $n$-variable monomials, and let $L$ be the Riesz functional associated to $s$. In previous work [arXiv:1804:04276, 2018] we used an iterative geometric construction to associate to $L$ a sequence of algebraic sets in $\mathbb{R}^{n}$, $S_{0} \supseteq S_{1} \supseteq \cdots$, such that $L$ has a representing measure if and only if the sequence stabilizes at a nonempty set $S_{k} = S_{k+1} = \cdots$, called the core variety of $L$, $CV(L)$. In this case, $CV(L)$ is precisely the union of supports of all finitely atomic representing measures for $L$. This result applies more generally when $L$ acts on a finite dimensional vector space of Borel measurable functions on a $T_{1}$ topological space. In the present work we extend the idea of the core variety so as to provide a test for membership in a prescribed convex cone. Let $T$ be a subset of a finite dimesnional real vector space $V$, and let $C$ be the conical hull of $T$. Given $L$ in $V$, we use an iterative geometric construction to define the core variety $CV(L)$ to be a subset of $T$. The main result is that $L$ belongs to $C$ if and only if the iterative procedure stabilizes at nonempty subset. In this case, $CV(L)$ consists of those elements of $T$ which appear in some representation of $L$ as a conical combination of elements of $T$.  

Jean Pierre Gabardo
McMaster University

A factorization problem related to the convolution of positive definite functions.

Let $G$ be one of the classical locally compact abelian groups: $\mathbb{Z}_N^d$, $\mathbb{Z}^d$, $\mathbb{T}^d$ or $\mathbb{R}^d$.

Suppose that  $U$ is an open subset of $G$ with finite Haar measure and that $U$ is symmetric in the sense that $0\in U$ and that $-x\in U$ whenever  $x\in U$. We will discuss the problem of factorizing the constant function $1$ on $G$ as the convolution of two positive definite objects on $G$, one being a continuous positive definite function which is identically zero outside of $U$ and the other a positive definite function (or distribution depending on the group) supported on the set $\{0\}\cup (G\setminus U)$. We will provide a sufficient condition for the problem to be solvable and present some applications of our result.

Didier Henrion
LAAS-CNRS; University of Toulouse

Tractable semi-algebraic approximation using Christoffel-Darboux kernel

We provide a new method to approximate a (possibly discontinuous) function using Christoffel-Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy solutions to non-linear hyperbolic PDEs, or using numerical integration from finitely many evaluations of the function. While most of the existing methods construct a piecewise polynomial approximation, we construct a semi-algebraic approximation whose estimation and evaluation can be performed efficiently. An appealing feature of this method is that it deals with non-smoothness implicitly so that a single scheme can be used to treat smooth or non-smooth functions without any prior knowledge. On the theoretical side, we prove pointwise convergence almost everywhere as well as convergence in $L^1$ under broad assumptions. Using more restrictive assumptions, we obtain explicit convergence rates. We illustrate our approach on various example from control and approximation. In particular we observe empirically that our method does not suffer from the the Gibbs phenomenon when approximating discontinuous functions.

Joint work with Jean Bernard Lasserre, Swann Marx, Edouard Pauwels and Tillmann Weisser.

David Kimsey
Newcastle University

On a minimal solution of the indefinite multidimensional truncated moment problem

Given a truncated multisequence $s$ and non-negative integers $\kappa_{\pm}$, we exhibit necessary and sufficient conditions for $s$ to have a representing measure $\mu = \mu_+ - \mu_-$, where ${\rm card}\, {\rm supp} \, \mu_{\pm} = \kappa_{\pm}$. We shall see that necessary and sufficient conditions can be formulated in terms of a rank preserving Hankel extension asuch that the polynomial ideal which models the aforementioned linear dependence is real radical. In the case that $\kappa_{-} = 0$, then our main result collapses to Curto and Fialkow's well-known flat extension theorem. The proof of the main result relies on some theory of Pontryagin spaces and also some basic results in commutative algebra.

Tobias Kuna
University of Reading

On truncated discrete moment problems

The discrete truncated moment problem considers the question whether given a discrete subsets $K \subset \mathbb{R}^n$ and a collection of real numbers one can find a measure supported on $K$ whose (power) moments (up to degree $d$) are exactly these numbers. In the case $n=1$ I will present  a minimal set of necessary and sufficient conditions for the existence of such measures. I will discuss where we stand in the case $ n \gt 1$, $K =\mathbb{Z}^n$ and $d=2$. This simple problem is surprisingly hard and not treatable with known techniques. Applications to the truncated moment problem for point processes, the so-called relizability or representability problem are given. This is a joint work with M. Infusino, J. Lebowitz and E. Speer.

David de Laat
TU Delft

Pair correlation estimates for the zeros of the zeta function via sum-of-squares representations

In this talk I will explain how sum-of-squares characterizations and semidefinite programming can be used to obtain improved bounds for quantities related to zeros of the Riemann zeta function. This is based on Montgomery's pair correlation approach. I will show how this connects to the sphere packing problem, and speculate about future improvements.

Joint work with Andrés Chirre and Felipe Gonçalves.

Swann Marx

A moment approach for entropy solutions to scalar conservation laws.

In this talk, we will propose a new numerical scheme, based on the so called Lasserre hierarchy, which solves scalar conservation laws. Our approach is based on a very weak notion of solution introduced by DiPerna in 1985, which is called entropy measure-valued solution. Among other nice properties, this formulation is linear in a Borel measure, which is the unknown of the equation, and moreover it is equivalent to the well-known entropy solution formulation. Our aim is to explain that the Lasserre hierarchy allows to solve such a linear equation without relying on a mesh, but rather by truncating the moments of the measure under consideration up to a certain degree.

Maria João Oliveira
Universidade Aberta; CMAFcIO, University of Lisbon

Bogoliubov generating functionals for interacting particle systems in the continuum

One presents an overview how correlation measures (factorial moment measures) and related generating functions can be used to study the time evolution of an interacting particle system in the continuum.

Rajae Ben Taher
University Moulay Ismail, Faculty of Science, Department of Mathematics, Laboratory of Mathematics and their Applications

A combinatorial expression for the $2$ variable moment sequence via Fibonacci sequence

In this talk, we are going to exploit the combinatorial expression of generalized Fibonacci sequences. More precisely, we manage to obtain the combinatorial expression for each term of the $2$-variable moment sequence, which admits a finitely atomic representing measure.

Martin Venker
Ruhr University Bochum

On the Random Moment Problem

The random moment problem is concerned with determining typical behavior of moment sequences by studying appropriate moment spaces equipped with probability distributions. We consider random moment sequences of probability measures on a subset $E$ of the real line in the three classical cases of $E$ being the unit interval, the half-line and the whole real line, respectively. We find, depending on $E$, the moments of the families of Kesten-McKay-, Marchenko-Pastur- and semicircle distribution being typical, respectively. Special emphasis is given to universality questions, i.e. the question how typical these three families are. Moreover, we determine typical moment sequences if some moments are known a priori.