The Moment Problem for Continuous Positive Semidefinite Linear functionals

Details The Moment Problem for Continuous Positive Semidefinite Linear functionals The Moment Problem for Continuous Positive Semidefinite Linear functionals

2010-10-14

arxiv.org/abs/1010.2796v4

Mathematics

Algebraic Geometry

21 pages

Scientific paper

Let $V$ be the countable dimensional polynomial $\reals$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $\tau$ be a locally convex topology on V. Let $K$ be a closed subset of $\reals^n$, and let $M:=M_{\{g_1, ... g_s\}}$ be a finitely generated quadratic module in $V$. We investigate the following question: When is the cone $\Pos(K)$ (of polynomials nonnegative on $K$) included in the closure of $M$? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of $M=\sos$ with respect to weighted norm-$p$ topologies. We show that this closure coincides with the cone $\Pos(K)$ where $K$ is a certain convex compact polyhedron. We use these results to generalize Berg's et al work on exponentially bounded moment sequences.

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