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

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 $\lambda=0.24565978 \ldots$ and $\theta=0.08899552\ldots$ 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 $2 + 2 \theta = 2.17799105\ldots$