For Lagrange polynomial interpolation on open arcs $X=\gamma$ in $\mathbb{C}$, it is well-known that the Lebesgue constant for the family of Chebyshev points ${\bf{x}}_n:=\{x_{n,j}\}^{n}_{j=0}$ on $[-1,1]\subset \mathbb{R}$ has growth order of $O(log(n))$. The same growth order was shown in [45] for the Lebesgue constant of the family ${\bf {z^{**}_n}}:=\{z_{n,j}^{**}\}^{n}_{j=0}$ of some properly adjusted Fej\'er points on a rectifiable smooth open arc $\gamma\subset \mathbb{C}$. On the other hand, in our recent work [15], it was observed that if the smooth open arc $\gamma$ is replaced by an $L$-shape arc $\gamma_0 \subset \mathbb{C}$ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz-Zygmund inequalities are no longer valid for the family of Fej\'er points ${\bf z}_n^{*}:=\{z_{n,j}^{*}\}^{n}_{j=0}$ on $\gamma$, and that the rate of growth for the corresponding Lebesgue constant $L_{{\bf {z}}^{*}_n}$ is as fast as $c\,log^2(n)$ for some constant $c>0$. The main objective of the present paper is 3-fold: firstly, it will be shown that for the special case of the $L$-shape arc $\gamma_0$ consisting of two line segments of the same length that meet at the angle of $\pi/2$, the growth rate of the Lebesgue constant $L_{{\bf {z}}_n^{*}}$ is at least as fast as $O(Log^2(n))$, with $\lim\sup \frac{L_{{\bf {z}}_n^{*}}}{log^2(n)} = \infty$; secondly, the corresponding (modified) Marcinkiewicz-Zygmund inequalities fail to hold; and thirdly, a proper adjustment ${\bf z}_n^{**}:=\{z_{n,j}^{**}\}^{n}_{j=0}$ of the Fej\'er points on $\gamma$ will be described to assure the growth rate of $L_{{\bf z}_n^{**}}$ to be exactly $O(Log^2(n))$.

翻译：对于复平面 $\mathbb{C}$ 中开弧 $X=\gamma$ 上的拉格朗日多项式插值，众所周知，Chebyshev 点族 ${\bf{x}}_n:=\{x_{n,j}\}^{n}_{j=0}$ 在 $[-1,1]\subset \mathbb{R}$ 上的 Lebesgue 常数具有 $O(log(n))$ 的增长阶。文献[45] 证明，对于可求长光滑开弧 $\gamma\subset \mathbb{C}$ 上适当调整的 Fejér 点族 ${\bf {z^{**}_n}}:=\{z_{n,j}^{**}\}^{n}_{j=0}$，其 Lebesgue 常数也呈现相同的增长阶。另一方面，在我们近期工作[15] 中发现，若将光滑开弧 $\gamma$ 替换为由两条线段构成的 $L$ 形弧 $\gamma_0 \subset \mathbb{C}$，数值实验表明 Marcinkiewicz-Zygmund 不等式对于 $\gamma$ 上的 Fejér 点族 ${\bf z}_n^{*}:=\{z_{n,j}^{*}\}^{n}_{j=0}$ 不再成立，且相应 Lebesgue 常数 $L_{{\bf {z}}^{*}_n}$ 的增长速率可达 $c\,log^2(n)$（其中 $c>0$ 为某常数）。本文的主要目标有三：首先，针对由两条等长线段以 $\pi/2$ 角相接构成的 $L$ 形弧 $\gamma_0$ 这一特例，证明 Lebesgue 常数 $L_{{\bf {z}}_n^{*}}$ 的增长速率至少为 $O(Log^2(n))$，且 $\lim\sup \frac{L_{{\bf {z}}_n^{*}}}{log^2(n)} = \infty$；其次，证明相应的（修正）Marcinkiewicz-Zygmund 不等式不成立；最后，将描述一种对 $\gamma$ 上 Fejér 点的适当调整 ${\bf z}_n^{**}:=\{z_{n,j}^{**}\}^{n}_{j=0}$，以确保 $L_{{\bf z}_n^{**}}$ 的增长速率精确为 $O(Log^2(n))$。