指数位于紧凸集中的多项式及相关加权极值函数——Bernstein-Walsh-Siciak定理
Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Bernstein-Walsh-Siciak theorem
AI总结：
本文将Bernstein-Walsh-Siciak定理推广至指数受限于紧凸集$mS$的多项式子环，并以容许权函数下的加权一致估计替代普通一致误差估计。
摘要：
我们将关于$\mathbb{C}^n$中多项式逼近的Bernstein-Walsh-Siciak定理推广到如下情形：多项式环$\mathcal{P}(\mathbb{C}^n)$被其子环$\mathcal{P}^S(\mathbb{C}^n)$取代，该子环由所有指数限制在集合$mS$中的多项式组成，其中$S$是$\mathbb{R}_+^n$的紧凸子集，满足$0\in S$且$m=0,1,2,3,\dots$；同时，逼近中的一致误差估计被关于容许权函数的加权一致估计所取代。
英文摘要：
We generalize the Bernstein-Walsh-Siciak theorem on polynomial approximation in $\mathbb{C}^n$ to the case where the polynomial ring $\mathcal{P}(\mathbb{C}^n)$ is replaced by a subring $\mathcal{P}^S(\mathbb{C}^n)$ consisting of all polynomials with exponents restricted to sets $mS$, where $S$ is a compact convex subset of $\mathbb{R}_+^n$ with $0 \in S$ and $m = 0, 1, 2, 3, \dots$, and uniform estimates of error in the approximation are replaced by weighted uniform estimates with respect to an admissible weight function.
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