Order of uniform approximation by polynomial interpolation in the complex plane and beyond
For Lagrange polynomial interpolation on open arcs $X=γ$ 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 Fejér points on a rectifiable smooth open arc $γ\subset \mathbb{C}$. On the other hand, in our recent work [15], it was observed that if the smooth open arc $γ$ is replaced by an $L$-shape arc $γ_0 \subset \mathbb{C}$ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz-Zygmund inequalities are no longer valid for the family of Fejér points ${\bf z}_n^{*}:=\{z_{n,j}^{*}\}^{n}_{j=0}$ on $γ$, and that the rate of growth for the corresponding Lebesgue constant $L_{\bf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 $γ_0$ consisting of two line segments of the same length that meet at the angle of $π/2$, 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 Fejér points on $γ$ will be described to assure the growth rate of $L_{{\bf z}_n^{**}}$ to be exactly $O(Log^2(n))$.