Improved polynomial estimate for the Lebesgue constants of Leja sequences on finite unions of intervals

Abstract

We prove a new polynomial upper bound for the Lebesgue constants of $\tau$-Leja sequences on finite unions of real intervals. Building on an estimate of Andrievskii and Nazarov, we replace the global separation of the first $n$ Leja points by a local separation estimate at the Green-function scale ${\rho }_{1/n}$. Combined with a packing argument and estimates on ${\rho }_{1/n}$ near and away from the endpoints, this yields ${\Lambda }_n=O(n^{2{\alpha }_{\tau }})$ uniformly over all possible $\tau$-Leja sequences, with ${\alpha }_{\tau }=1+\theta +2{\lambda }^{-1}\ln ({\tau }^{-1})$, where $\lambda =0.24565978\dots$ and $\theta =0.08899552\dots$ In particular, for genuine Leja sequences on finite unions of intervals, including the benchmark case $K=[-1,1]$, this improves the previously known best exponent $13/4=3.25$ to around $2+2\theta =2.17799105\dots$