AI 中文总结
研究亏格\(g>0\)紧致黎曼曲面上切比雪夫极值问题，考虑特定容许亚纯函数，证明相关常数收敛及不等式。利用柯西核研究维德姆因子渐近性，揭示无类似法伯型渐近性，最后在亏格\(1\)时构造了明确例子。
AI 中文摘要
我们研究亏格\(g>0\)的紧致黎曼曲面\(X\)上的切比雪夫极值问题。作为首一多项式的类似物，我们考虑在标记点\(P_{\infty}\)之外无极点（并进行适当归一化）的容许亚纯函数。对于非极紧致集\(E\subset X\setminus\{P_{\infty}\}\)，我们证明切比雪夫常数\(t_n(E)\)的\(n\)次方根收敛到\(E\)的容量，并建立相应的伯恩斯坦 - 沃尔什不等式。在论文第二部分，利用适应黎曼曲面的柯西核，我们研究极值函数的精细塞戈 - 维德姆渐近性以及维德姆因子\(W_n(E)=\frac{t_n(E)}{\text{cap}(E)^n}\)，假设\(E\)是\(p\)个具有解析边界的闭圆盘的有限并集。亏格为\(2g + p - 1\)的肖特基双的几何结构明确地进入这些渐近性中。我们发现即使对于一条边界曲线也没有法伯型渐近性的类似物。最后我们使用魏尔斯特拉斯\(\wp\)函数在亏格为\(1\)的情况下构造了明确的例子。
英文摘要
We study the Chebyshev extremal problem on a compact Riemann surface $X$ of genus $g>0$. As an analog to monic polynomials, we consider admissible meromorphic functions having no poles away from a marked point $P_{\infty}$ (with adequate normalisation). For a nonpolar compact set $E\subset X\setminus\{P_{\infty}\}$, we show that the $n$-th root of the Chebyshev constant $t_n(E)$ converges to the capacity of $E$, and we establish the corresponding Bernstein-Walsh inequality. In the second part of the paper, using a Cauchy kernel adapted to Riemann surfaces, we study the refined Szegő--Widom asymptotics for the extremals as well as the Widom factors $$ W_n(E)=\frac{t_n(E)}{\operatorname{cap}(E)^n},$$ assuming that $E$ is a finite union of $p$ closed discs with analytic boundaries. The geometry of the Schottky double, which has genus $2g+p-1$, enters explicitly into these asymptotics. We find that there is no analogue of Faber-type asymptotics even for one single boundary curve. We conclude by constructing explicit examples in genus $1$ using the Weierstrass-$\wp$ function.
Comments63 pages
