发表机构
Université Paris Cité, CNRS, MAP5（巴黎西岱大学，法国国家科学研究中心，MAP5）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
通过局部分离估计和打包论证，将Leja序列的Lebesgue常数上界改进为O(n^{2α_τ})，在标准区间[-1,1]上指数从13/4降至约2.178。
AI 中文摘要
我们证明了实区间有限并集上$\tau$-Leja序列的Lebesgue常数的一个新的多项式上界。基于Andrievskii和Nazarov的估计，我们将前$n$个Leja点的全局分离替换为Green函数尺度$\rho_{1/n}$下的局部分离估计。结合打包论证以及$\rho_{1/n}$在端点附近和远离端点处的估计，这统一地给出了对所有可能的$\tau$-Leja序列有$\Lambda_n = O(n^{2\alpha_\tau})$，其中$\alpha_\tau = 1+\theta+2\lambda^{-1}\ln(\tau^{-1})$，$\lambda=0.24565978 \ldots$，$\theta=0.08899552\ldots$。特别地，对于区间有限并集上的真正Leja序列，包括基准情形$K = [-1,1]$，这将先前已知的最佳指数$13/4 = 3.25$改进到约$2 + 2 \theta = 2.17799105\ldots$。
英文摘要
We prove a general lower bound for the Lebesgue constants $Λ_n$ of Leja sequences for every nonpolar compact set \(K\subset\mathbb C\): $\textstyle \limsup_{n \to+\infty} Λ_n \, \sqrt{n^{-1} W_n(K)}\geq \tfrac{1}{\sqrt{2}}$, where \(W_n(K)\) denotes the Chebyshev--Widom factor. In particular, bounded Widom factors imply a universal square root obstruction. We also establish improved polynomial upper bounds on compact subsets $K$ of the real line, building on the approach of Andrievskii and Nazarov. Our first ingredient is a refinement of their Key Lemma: we lower its exponent from $9/8 = 1.125$ to a near-optimal $γ^\star \approx 0.6304$. Our second ingredient is a localisation of the separation argument: instead of using the global minimum spacing among the first $n$ Leja points, we retain the local Green-function scale. For finite unions of intervals, a packing argument and sharp estimates on the latter function near and away from the endpoints yield $Λ_n = O(n^{1 + γ^\star})$, uniformly over Leja sequences. This improves the previously known best exponent $13/4 = 3.25$ to around $1.6304$ for such sets (including the benchmark case $K = [-1,1]$), hence substantially narrowing the gap with the near-linear growth observed numerically. Corresponding polynomial bounds for $τ$-Leja sequences are derived.
