The Multidimensional Moment Problem: Determinacy

Abstract

In this chapter, we study the determinacy problem in the multivariate case. In Sect. 14.1, we introduce several natural determinacy notions (strict determinacy, strong determinacy, ultradeterminacy) that are all equivalent to the “usual” determinacy in dimension one. In the remaining sections we develop various techniques and methods to derive sufficient criteria for determinacy. In Sect. 14.2, polynomial approximation is used to show that the determinacy of all marginal sequences implies the determinacy of a moment sequence (Theorem 14.6). Section 14.3 is based on operator-theoretic methods in Hilbert space. The main results (Theorems 14.12 and 14.16) show that the determinacy of appropriate 1-subsequences of a positive semidefinite d-sequence s implies that s is a (determinate) moment sequence. Section 14.4 is concerned with Carleman’s condition in the multivariate case. Probably the most useful result in this chapter is Theorem 14.20; it says that if all marginal sequences of a positive semidefinite d-sequence s satisfy Carleman’s condition, then s is a determinate moment sequence. Section 14.6 uses the disintegration of measures as a powerful method for the study of determinacy. A fibre theorem for determinacy (Theorem 14.30) states a measure is determinate if the base measure is strictly determinate and almost all fibre measures are determinate. In Sect. 14.5, we calculate the moments of the surface measure on S ^d−1 and of the Gaussian measure on $\mathbb{R}^{d}$.