Order of uniform approximation by polynomial interpolation in the complex plane and beyond

Abstract

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 ${\mathbf{x}}_n:=\{x_{n,j}{\}}_{j=0}^n$ 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 ${\mathbf{z}}_{\mathbf{n}}^{\ast \ast }:=\{z_{n,j}^{\ast \ast }{\}}_{j=0}^n$ of some properly adjusted Fejér 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consisting of two line segments, numerical experiments suggest that the Marcinkiewicz-Zygmund inequalities are no longer valid for the family of Fejér points ${\mathbf{z}}_n^{\ast }:=\{z_{n,j}^{\ast }{\}}_{j=0}^n$ on $\gamma$, and that the rate of growth for the corresponding Lebesgue constant $L_{{\mathbf{z}}_{\mathbf{n}}^{\ast }}$ is as fast as $clo{g}^2(n)$ 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 $L_{{\mathbf{z}}_{\mathbf{n}}^{\ast }}$ is at least as fast as $O(Lo{g}^2(n))$, with $\lim \sup \frac{L_{{\mathbf{z}}_{\mathbf{n}}^{\ast }}}{lo{g}^2(n)}=\infty$; secondly, the corresponding (modified) Marcinkiewicz-Zygmund inequalities fail to hold; and thirdly, a proper adjustment ${\mathbf{z}}_n^{\ast \ast }:=\{z_{n,j}^{\ast \ast }{\}}_{j=0}^n$ of the Fejér points on $\gamma$ will be described to assure the growth rate of $L_{{\mathbf{z}}_n^{\ast \ast }}$ to be exactly $O(Lo{g}^2(n))$.

Submit to Indagationes Mathematicae, Prof. Jaap Korevaar 100-th birthday special issue, 32 pages, no figures, keywords:Lebesgue constants; Marcinkiewicz-Zygmund inequalities; Fejer points; Conformal Mapping