RUI: Truncated Multivariable Moment Problems & Applications: An Operator Theoretic Approach

Suny College At New Paltz, New Paltz NY

Investigators

Abstract

ABSTRACT This research concerns an operator-theoretic approach to multivariable moment problems. The prototypical problem that we study is the Multivariable Truncated Moment Problem: Given a finite d-dimensional real sequence, we seek concrete necessary and sufficient conditions so that there exists a positive Borel measure on d-dimensional Euclidean space for which the given data represent successive power moments of the measure. To study the existence of such a representing measure, we associate to the data a finite moment matrix. It is known that a finitely atomic representing measure exists if and only if the moment matrix admits an extension to an infinite, finite-rank, positive moment matrix. Representing measures with the fewest atoms correspond to extensions of minimal rank. We seek to establish concrete necessary and sufficient conditions for such extensions, and also to develop algorithms for explicitly computing representing measures corresponding to extensions. In the case when the moment matrix is singular, we study the following conjecture: there exists a representing measure if and only if the moment matrix is positive, recursively generated, and the rank of the matrix is at most equal to the size of the algebraic variety natuarally associated to the data. This conjecture is true for moment problems on planar curves of degree one or two, so we study the conjecture for curves of higher degree. This research also concerns estimates of the minimal rank in the above-mentioned extensions; such estimates are related to the convergence of certain polynomial optimization algorithms and also to the size of minimal cubature rules in Numerical Analysis. The aim of this research is to develop new existence and uniqueness criteria for finitely atomic representing measures in multivariable truncated moment problems (with data corresponding to successive power moments up to a fixed finite degree). Truncated moment problems play essential roles in aspects of such fields as Operator Theory (subnormality of weighted shifts), Interpolation Theory (classical Nevanlinna-Pick theory), Numerical Analysis (multivariable cubature rules), Control Theory (signal processing), and Optimization Theory (polynomial optimization over a region). The principal focus of this research is an approach to multivariable truncated moment problems based on an extension theory for the moment matrix associated to the moment data. When this matrix admits an infinite, positive, finite rank moment matrix extension, this approach yields an explicit formula for a finitely atomic representing measure. The primary goal of this research is to determine concrete criteria on the moment data which permit the desired extension. This research also concerns the development of algorithms to implement these criteria. One principal application will be to develop new minimal cubature rules for measures on classical domains such as the disk and triangle; another application concerns the convergence of polynomial optimization algorithms. Broader impacts will include undergraduate training and research projects for science students from underrepresented minorities, and the use of computing, particularly simulations, as an experimental methodology in mathematics and computer science courses.