论文标题
最佳函数函数通过对数多项式的效果
Best approximation of functions by log-polynomials
论文作者
论文摘要
Lasserre [la]证明，对于每个紧凑型设置$ k \ subset \ mathbb r^n $，每个偶数$ d $都存在一个唯一的同质多项式$ g_0 $ g_0 $ d $ $ d $，带有$ k \ subset g_1（g_0）（g_0） $ | g_1（g）| $在所有此类多项式中，$ g $满足条件$ k \ subset g_1（g）$。该结果扩展了Löwner椭圆形的概念，不仅从凸体到任意的紧凑型集（如果$ d = 2 $，则通过拿到凸面船体即时），而且还从椭圆形到任意均匀程度的均质多项式的水平集。 在本文中，我们将此结果扩展到以两种不同方式的非阴性对数concove函数的类别。其中一个是已知结果的直接扩展，另一个是合适的扩展，在相应的问题中具有唯一的溶液，并且在某些“接触点”方面进行了表征。
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'.
