AI 中文总结
本文研究实轴半正则子集上切比雪夫与剩余多项式的Widom因子，推导其上下界，明确其有界性条件，在特殊情形下得到精确极限与Szegő–Widom渐近式。
AI 中文摘要
我们研究实轴上紧非极小子集上切比雪夫多项式与剩余多项式的L∞ Widom因子，这类子集不必满足位势理论意义下的正则性，尤其包含带孤立点的集合。首先证明Widom因子的有界性与归一化点x*∈R̄\backslashE无关。对于正则点集E^reg为闭集的半正则集，我们引入非正则系数IR(E,x*)=∑_{x∈E\backslashE^reg}G_E(x,x*)，用于度量非正则边界点对极值多项式问题的贡献。我们得到上界：sup_{n≥1}W_n(E,x*)≤2exp[PW(E^reg,x*)+IR(E,x*)]，互补下界为：liminf_{n→∞}W_n(E,x*)≥2exp[IR(E,x*)]。因此，对于半正则Parreau–Widom集，Widom因子有界当且仅当IR(E,x*)<∞。当正则部分为有限区间的并时，该条件等价于非正则点满足几何平方根可和性条件。在E=[a,b]∪{x_k}_{k≥1}的特殊情形下，Widom因子具有精确的、可能为无穷的极限2exp[IR(E,x*)]。当该极限有限时，我们得到归一化极值多项式对应的Szegő–Widom渐近式。
英文摘要
We study $L^\infty$ Widom factors for Chebyshev and residual polynomials on compact non-polar subsets of the real line that need not be regular in the sense of potential theory and, in particular, on sets with isolated points. We first show that boundedness of the Widom factors is independent of the normalization point $x_*\in\overline{\mathbb{R}}\backslash\mathsf{E}$. For semi-regular sets, meaning that the set of regular points $\mathsf{E}^\mathrm{reg}$ is closed, we introduce the irregularity coefficient $\mathcal{IR}(\mathsf{E},x_*)=\sum_{x\in\mathsf{E}\backslash\mathsf{E}^\mathrm{reg}}G_\mathsf{E}(x,x_*)$, which measures the contribution of irregular boundary points to the extremal polynomial problem. Our upper bound is \[ \sup_{n\ge1}\mathcal{W}_n(\mathsf{E},x_*) \le 2\exp\bigl[\mathcal{PW}(\mathsf{E}^\mathrm{reg},x_*)+\mathcal{IR}(\mathsf{E},x_*)\bigr], \] while the complementary lower bound is \[ \liminf_{n\to\infty}\mathcal{W}_n(\mathsf{E},x_*) \ge 2\exp\bigl[\mathcal{IR}(\mathsf{E},x_*)\bigr]. \] Consequently, for semi-regular Parreau--Widom sets the Widom factors are bounded if and only if $\mathcal{IR}(\mathsf{E},x_*)<\infty$. When the regular part is a finite union of intervals, this condition is equivalent to a geometric square-root summability condition on the irregular points. In the special case $\mathsf{E}=[a,b]\cup\{x_k\}_{k\ge1}$, the Widom factors have the exact, possibly infinite, limit $2\exp[\mathcal{IR}(\mathsf{E},x_*)]$. When this limit is finite, we obtain the corresponding Szegő--Widom asymptotics for the normalized extremal polynomials.
Comments21 pages
