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

Suny College At New Paltz, New Paltz NY

Investigators

Abstract

Truncated Multivariable Moment Problems and Applications: An Operator Theoretic Approach This research concerns an operator-theoretic approach to multivariable truncated moment problems. For a finite real multi-sequence, we seek concrete necessary and sufficient conditions for the existence of a positive Borel representing measure in Euclidean space, a measure such that the power moments of the measure coincide with the corresponding elements of the multi-sequence. We associate to the multi-sequence a corresponding moment matrix. It is known that a representing measure exists if and only if the moment matrix admits an extension to a larger, positive semi-definite moment matrix, which in turn admits a flat, i.e., rank-preserving, extension to a still larger moment matrix. This research concerns the existence and minimal size of such moment matrix extensions. Column dependence relations in the moment matrix determine an algebraic variety which contains the support of any representing measure, and this research concerns a description of the algebraic varieties for which it is possible to establish the desired flat moment matrix extensions. More generally, this research concerns representing measure whose support is contained in a prescribed semi-algebraic closed set. In this case, we require flat extensions (as above) for which the localizing matrices corresponding to the semi-algebraic set are also positive semi-definite. This part of the research is concerned with semi-algebraic (or algebraic) sets for which positive polynomials admit degree-bounded weighted sum-of-squares representations; applications directly concern finite convergence in Lasserre's polynomial optimization theory. Another aspect of this research concerns algorithms for explicitly computing finitely atomic representing measures; applications of this part of the research lead to the construction of minimal or near-minimal multivariable 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. Truncated moment problems play an essential role 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), Optimization Theory (polynomial optimization over a region), and Real Algebraic Geometry (representations of positive polynomials as weighted sums-of-squares). The principal focus of this research is an approach to multidimensional 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 supported on the joint spectrum of a normal tuple of operators corresponding to the extension. The primary goal of this research is to determine concrete conditions on the moment data which permit the desired extension. This research also concerns algorithms for explicitly computing finitely atomic representing measures corresponding to moment matrix extensions. Another aspect of this research concerns the interplay between the existence of representing measures and the existence of sum-of-squares representations for positive polynomials; this aspect is directly related to Lasserre's algorithm for polynomial optimization. Another application of this research concerns the development of new minimal cubature rules on classical domains such as the disk or triangle. 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.