The K-moment problem with densities

• 成果类型: Article; Proceedings Paper
• 署名作者: Lasserre, Jean B.
• 署名单位: Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-007-0118-4
• 发表日期: 2009
• 页码: 321-341
• 关键词: Polynomials

摘要

Given a compact basic semi-algebraic set K subset of R-n, a rational fraction f : R-n -> R, and a sequence of scalars y = (y(alpha)), we investigate when y(alpha) = integral K x(alpha) f d mu holds for all alpha is an element of N-n, i.e., when y is the moment sequence of some measure f d mu, supported on K. This yields a set of (convex) linear matrix inequalities(LMI). We also use semidefinite programming to develop a converging approximation scheme to evaluate the integral integral f d mu when the moments of mu are known and f is a given rational fraction. Numerical expreriments are also provided. We finally provide (again LMI) conditions on the moments of two measures v, mu with support contained in K, to have dv = f d mu for some rational fraction f.