实数子集上L^p极值多项式的四向Szegő定理
A four-way Szegő theorem for $L^p$ extremal polynomials on subsets of $\mathbb R$
浏览论文内容
中文总结 AI 辅助
本文针对实数上特定紧支撑集的L^p极值多项式证明四向Szegő定理，推导四个条件的等价性、Widom因子的上下界，并给出实例说明结果的尖锐性。
中文摘要 AI 辅助
我们针对实数上具有紧支撑集K=K₀∪X的L^p极值多项式证明了一个四向Szegő定理，其中K₀是正则紧集，X是有限或可数个孤立点的集合。对于满足权重相应假设的2≤p≤∞（包括加权切比雪夫情形p=∞），Parreau–Widom关于K₀的条件、X的Blaschke条件、权重的Szegő条件以及Widom条件0<limsup_{n→∞}W_{p,n}<∞这四个条件中，任意三个都能推出第四个。由此可得，对于实数上每个具有正容量的正则紧集K，Parreau–Widom条件等价于无权重切比雪夫Widom因子的有界性，也等价于平衡测度L² Widom因子的有界性。对于每个0<p≤∞，我们还证明了Widom因子的上下界，其中权重、孤立点和K₀的间隙的贡献分别呈现。最后，我们给出例子说明所得结果的尖锐性。对于p=2，我们实现了本工作所证明的蕴含关系未排除的以下五个性质的所有组合：Parreau–Widom条件、Blaschke条件、Szegő条件、Widom因子的上有界性、Widom因子远离零的有界性。
英文摘要
We prove a four-way Szegő theorem for $L^p$ extremal polynomials on compact supports $K=K_0\cup X\subset\mathbb R$, where $K_0$ is a regular compact set and $X$ is a finite or countable set of isolated points. For every $2\le p\le\infty$, including the weighted Chebyshev case $p=\infty$ under the corresponding assumptions on the weight, any three of the Parreau--Widom condition for $K_0$, the Blaschke condition for $X$, the Szegő condition for the weight, and the Widom condition $0<\limsup_{n\rightarrow\infty}W_{p,n}<\infty$ imply the fourth. As a consequence, for every regular compact set $K\subset\mathbb R$ of positive capacity, the Parreau--Widom condition is equivalent both to boundedness of the unweighted Chebyshev Widom factors and to boundedness of the equilibrium-measure $L^2$ Widom factors. For every $0<p\le\infty$, we also prove upper and lower bounds for the Widom factors in which the contributions of the weight, the isolated points, and the gaps of $K_0$ appear separately. Finally, we give examples illustrating the sharpness of our results. For $p=2$, we realize every combination of the following five properties that is not excluded by the implications proved in this work: the Parreau--Widom condition, the Blaschke condition, the Szegő condition, boundedness of the Widom factors from above, and boundedness of the Widom factors away from zero.
