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

Suny College At New Paltz, New Paltz NY

Investigators

Abstract

PI: Lawrence A. Fialkow, SUNY New Paltz DMS-0201430 Abstract This research concerns an approach to the multidimensional truncated power moment problem based on an extension theory for the associated moment matrix. When this matrix admits an infinite, positive, finite rank moment matrix extension, this approach yields an explicit formula for a finitely atomic representing measure supported on the joint spectrum of a normal tuple of operators corresponding to the extension. The aim of this research is to determine concrete conditions on the moment data which permit the desired extension. Much of this research concerns, specifically, the multivariable truncated K-moment problem, where the support of a representing measure is required to be contained in a prescribed closed set K. For K an algebraic variety, this research concerns a new conjecture for solving the moment problem in terms of concrete algebraic and geometric invariants closely related to the moment data. For K semi-algebraic, this research seeks to apply the abstract solution of the truncated K-moment problem due to Curto-Fialkow to specific semi-algebraic sets such as the closed unit disk. A direct application of this study concerns the multidimensional cubature problem in Numerical Analysis. By applying the moment matrix extension technique in the context of cubature, this research seeks to construct minimal cubature rules for sets such as the disk, square, triangle, or annulus. One aspect of this research, the Quadrature Problem, concerns the efficient measurement of the size of an irregular area, or the measurement of the weight of a volume whose density is unevenly distributed. We seek to identify a small number of test points (or nodes) within the body in such a way that by measuring the density just at these few points, we may closely approximate the overall size or weight of the body. In the case of a linear body, such as a thin rod, the quadrature problem was solved with the fewest number of test points by the mathematician Carl Friedrich Gauss (1777-1855), using a technique now known as Gaussian Quadrature. At the present time, surprisingly few minimal-node quadrature rules are known for shapes in the plane or in 3-dimensional space, even for basic sets such as a disk or sphere. In order to study the Quadrature Problem, we actually study a more general problem, the Multidimensional Truncated Moment Problem. Many real-world systems can be described by a sequence of physical attributes called moments, such as mass, weight, momentum, etc., which relate to the physical space underlying the system. The truncated moment problem asks whether a system is uniquely determined by its sequence of moments, and also whether the moments of a system can be computed by studying the system at just a finite number of nodes in the underlying space. In this research, we develop algorithms for recognizing when such a sequence of nodes exists, and for efficiently computing them. In this way, we may describe a system in terms of just a finite number of points in the underlying space.