对数多项式对函数的最佳逼近
Best approximation of functions by log-polynomials
摘要：
Lasserre [La]证明了对于每个紧集$K\subset\mathbb R^n$和每个偶数$d$，存在唯一的齐次多项式$g_0$的次数$d$与$K\subset G_1(g_0)=\{x\in\mathbb R^n:g_0(x)\leq 1\}$最小化$|G_1(g)|$在所有这样的多项式$g$满足条件$K\subset G_1(g)$ .这个结果扩展了Löwner椭球的概念，不仅从凸体到任意紧集（如果$d=2$采用凸包，这是直接的），而且从椭球到任意偶数次齐次多项式的水平集。本文用两种不同的方法将这一结果推广到非负对数凹函数类。其中之一是已知结果的直接扩展，另一个是一个适当的扩展，在相应的问题的解决方案的唯一性和一些“接触点”的特征。
英文摘要：
Lasserre [La] proved that for every compact set $K\subset\mathbb R^n$ and every even number $d$ there exists a unique homogeneous polynomial $g_0$ of degree $d$ with $K\subset G_1(g_0)=\{x\in\mathbb R^n:g_0(x)\leq 1\}$ minimizing $|G_1(g)|$ among all such polynomials $g$ fulfilling the condition $K\subset G_1(g)$. This result extends the notion of the Löwner ellipsoid, not only from convex bodies to arbitrary compact sets (which was immediate if $d=2$ by taking convex hulls), but also from ellipsoids to level sets of homogeneous polynomial of an arbitrary even degree. In this paper we extend this result for the class of non-negative log-concave functions in two different ways. One of them is the straightforward extension of the known results, and the other one is a suitable extension with uniqueness of the solution in the corresponding problem and a characterization in terms of some 'contact points'.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
