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

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 $$G(\delta ):=\max \{g(z):z\in \mathbb{C},\mathrm{dist}(z,K)\le 2\delta \}.$$ Let $\{x_n{\}}_{n=0}^{\infty }$ be a Leja sequence of points of $K$. Then the uniform norm $\Vert T_n\Vert ={\Lambda }_n,n=1,2,\dots$ of the associated interpolation operator $T_n$, i.e., the $n$-th Lebesgue constant, is bounded from above by $$\underset{\delta >0}{\min }2n{\left[\frac{\mathrm{diam}(K)}{\delta }{e}^{nG(\delta )}\right]}^{9/8}.$$ 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.

arXiv comment19 pages, 2 figures