A simple upper bound for Lebesgue constants associated with Leja points on the real line
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 $δ>0$, define $$ G(δ):=\max\{ g(z): z\in \mathbb C, \,\operatorname{dist}(z,K)\le 2δ\}. $$ Let $\{ x_n\}_{n=0}^\infty$ be a Leja sequence of points of $K$. Then the uniform norm $\|T_n\|=Λ_n, n=1,2,\ldots$ of the associated interpolation operator $T_n$, i.e., the $n$-th Lebesgue constant, is bounded from above by $$ \min_{δ>0}2n\left[\frac{\operatorname{diam}( K)}δe^{nG(δ)}\right]^{9/8}. $$ 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.