Solving Moment Problems by Dimensional Extension

Abstract

polynomial inequalities. The second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part, for decomposing a polynomial which is positive on such a semi-algebraic set into a canonical sum of squares of rational functions times explicit multipliers. Let n‚ 1 be a flxed integer. Due to the fact that for n> 1 not every nonnegative polynomial in R n can be written as a sum of squares of polynomials (see, for instance, [2,x6.3]), the moment problems in n variables are more di‐cult than the classical one variable problems. This very intriguing territory has been investigated by many authors (see [2], [7], [12] and their references), although characterizations for measures whose support lies in an arbitrary (generally unbounded) semi-algebraic set do not seem to exist. The present paper starts from an idea of the second author, see [19], about solving moment problems by a change of basis via an embedding of R n into a submanifold of a higher dimensional Euclidean space. Rougly speaking we prove that certain (n + 1)-dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover, these extensions parametrize all possible solutions of the moment problem. To be more speciflc, let ∞fi = Z R n x fi d„(x) ;fi 2 Z n