Riesz泛函的正性与二次和四次矩问题的解
Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems
摘要：
我们利用Riesz泛函的正性来建立截断多元矩序列的表示测度（或近似表示测度）。对于截断矩序列$y$，我们表明，$y$在于关闭截断矩序列承认代表措施支持在一个规定的封闭集$K \subseteq \re^n$当且仅当相关的Riesz功能$L_y$是$K$ -积极的。对于一个确定集$K$，我们证明了如果$L_y$是严格$K$ -正的，则$y$允许一个支持于$K$的表示测度.因此，我们能够解决截断$K$ -矩问题的程度$k$的情况下：（i）$(n,k)=(2,4)$和$K=\re^2$ ;（ii）$n\geq 1$，$k=2$，和$K$是由一个二次等式或不等式定义。特别是，这些结果解决了截断矩问题的希尔伯特定理平方和的其余开放的情况下。
英文摘要：
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 $K \subseteq \re^n$ 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 $K=\re^2$; (ii) $n\geq 1$, $k=2$, 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.
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