Journal of Operator Theory

Volume 48, Issue 2, Fall 2002 pp. 315-354.

Solution of the singular quartic moment problem

Authors: Raul E. Curto (1), and Lawrence A. Fialkow (2)
Author institution: (1) Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242, USA
(2) Department of Mathematics and Computer Science, State University of New York, New Paltz, NY 12561, USA

Summary: In this note we obtain a complete solution to the quartic problem in the case when the associated moment matrix $M(2)(\gamma )$ is singular. Each representing measure $\mu $ satisfies $\mathop{\rm card}\mathop{\rm supp}\mu \geq \mathop{\rm rank}M( 2) $, and we develop concrete necessary and sufficient conditions for the existence and uniqueness of representing measures, particularly $\textit{minimal}$ ones. We show that $\mathop{\rm rank}% M( 2) $-atomic minimal representing measures exist in case the moment problem is subordinate to an ellipse or non-degenerate hyperbola. If the quartic moment problem is subordinate to a pair of intersecting lines, those problems subordinate to a general intersection of two conics may not have any representing measure at all. As an application, we describe the minimal quadrature rules of degree $4$ for arclength measure on a parabolic arc.

Keywords: Quartic moment problem, moment matrix extension, flat extensions, degree-one transformations, quadrature rules

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