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

TL;DR

This paper investigates the growth of Lebesgue constants for polynomial interpolation on complex plane arcs, showing that for certain shapes, the constants grow as fast as log-squared, and proposes adjustments to control this growth.

Contribution

It demonstrates that for L-shaped arcs, Lebesgue constants grow at least as fast as log-squared, and introduces a method to modify Fejér points to achieve controlled growth.

Findings

Lebesgue constants grow as fast as log^2(n) on L-shaped arcs

Marcinkiewicz-Zygmund inequalities fail for certain point sets

Adjusted Fejér points can ensure controlled growth of Lebesgue constants

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 ${\bf{x}}_n:=\{x_{n,j}\}^{n}_{j=0}$ 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consisting of two line segments, numerical experiments suggest that the Marcinkiewicz-Zygmund inequalities are no longer valid for the family of Fej\'er points ${\bf z}_n^{*}:=\{z_{n,j}^{*}\}^{n}_{j=0}$ on $\gamma$, and that the rate of growth for the corresponding Lebesgue constant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 $O(Log^2(n))$.