Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems

We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence $y$, we show that $y$ lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set $K\subseteq {\text{re}}^n$ if and only if the associated Riesz functional $L_y$ is $K$-positive. For a determining set $K$, we prove that if $L_y$ is strictly $K$-positive, then $y$ admits a representing measure supported in $K$. As a consequence, we are able to solve the truncated $K$-moment problem of degree $k$ in the cases: (i) $(n,k)=(2,4)$ and $K={\text{re}}^2$; (ii) $n\ge 1$, $k=2$, and $K$ is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert’s theorem on sums of squares.