Non-extremal sextic moment problems

Abstract

Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems. Also, positive semidefiniteness, combined with another necessary condition, consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r ) and the cardinality of the associated algebraic variety (denoted by v ) are equal. However, these conditions are not sufficient for non-extremal sextic or higher-order truncated moment problems. In this paper we settle three key instances of the non -extremal (i.e., r < v ) sextic moment problem, as follows: when r = 7 , positive semidefiniteness, consistency and the variety condition guarantee the existence of a 7-atomic representing measure; when r = 8 we construct two determining algorithms, corresponding to the cases v = 9 and v = + ∞ . To accomplish this, we generalize the rank-reduction technique developed in previous work, where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure.