A simple upper bound for Lebesgue constants associated with Leja points on the real line

TL;DR

This paper establishes a new upper bound for Lebesgue constants associated with Leja points on the real line, showing they grow at most polynomially for uniformly perfect sets, which is a novel result even for finite unions of intervals.

Contribution

The paper introduces a novel upper bound for Lebesgue constants of Leja points on the real line, applicable to general regular compact sets, including finite unions of intervals.

Findings

Lebesgue constants are bounded by a specific function involving the Green function.

For uniformly perfect sets, Lebesgue constants grow at most polynomially with n.

The result is new even for finite unions of intervals.

Abstract

Let $K\subset \mathbb R$ be a regular compact set and let $g(z)=g_{\overline{\mathbb C}\setminus K}(z,\infty)$ be the Green function for $\overline{\mathbb C}\setminus K$ with pole at infinity. For $\delta>0$, define $$ G(\delta):=\max\{ g(z): z\in \mathbb C, \,\operatorname{dist}(z,K)\le 2\delta\}. $$ Let $\{ x_n\}_{n=0}^\infty$ be a Leja sequence of points of $K$. Then the uniform norm $\|T_n\|=\Lambda_n, n=1,2,\ldots$ of the associated interpolation operator $T_n$, i.e., the $n$-th Lebesgue constant, is bounded from above by $$ \min_{\delta>0}2n\left[\frac{\operatorname{diam}( K)}{\delta}e^{nG(\delta)}\right]^{9/8}. $$ In particular, when $K$ is a uniformly perfect subset of $\mathbb R$, the Lebesgue constants grow at most polynomially in $n$. To the best of our knowledge, the result is new even when $K$ is a finite union of intervals.