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By Jean B. Lasserre


Abstract. Given a closed (and not necessarily compact) basic semi-algebraic set K ⊆ Rn,wesolvetheK-moment problem for continuous linear functionals. Namely,weintroduceaweightedℓ1-norm ℓw on R[x], and show that the ℓw-closures of the preordering P and quadratic module Q (associated with the generators of K) is the cone Psd(K) of polynomials nonnegative on K. We also prove that P and Q solve the K-moment problem for ℓw-continuous linear functionals and completely characterize those ℓw-continuous linear functionals nonnegative on P and Q (hence on Psd(K)). When K has a nonempty interior, we also provide in explicit form a canonical ℓw-projection gw f for any polynomial f, on the (degree-truncated) preordering or quadratic module. Remarkably, the support of gw f is very sparse and does not depend on K! This enables us to provide an explicit Positivstellensatz on K. And last but not least, we provide a simple characterization of polynomials nonnegative on K, which is crucial in proving the above results. 1

Year: 2013
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