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 $τ$-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 $ρ_{1/n}$. Combined with a packing argument and estimates on $ρ_{1/n}$ near and away from the endpoints, this yields $Λ_n = O(n^{2α_τ})$ uniformly over all possible $τ$-Leja sequences, with $α_τ= 1+θ+2λ^{-1}\ln(τ^{-1})$, where $λ=0.24565978 \ldots$ and $θ=0.08899552\ldots$ 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 $2 + 2 θ= 2.17799105\ldots$