Lebesgue constants of Leja sequences: a universal lower bound and improved upper bounds on the real line
Year: 2026
Type: preprint
Abstract: We prove a general lower bound for the Lebesgue constants $Λ_n$ of Leja sequences for every nonpolar compact set \(K\subset\mathbb C\): $\textstyle \limsup_{n \to+\infty} Λ_n \, \sqrt{n^{-1} W_n(K)}\geq \tfrac{1}{\sqrt{2}}$, where \(W_n(K)\) denotes the Chebyshev--Widom factor. In particular, bounded Widom factors imply a universal square root obstruction. We also establish improved polynomial upper bounds on compact subsets $K$ of the real line, building on the approach of Andrievskii and Nazarov. Our first ingredient is a refinement of their Key Lemma: we lower its exponent from $9/8 = 1.125$ to a near-optimal $γ^\star \approx 0.6304$. Our second ingredient is a localisation of the separation argument: instead of using the global minimum spacing among the first $n$ Leja points, we retain the local Green-function scale. For finite unions of intervals, a packing argument and sharp estimates on the latter function near and away from the endpoints yield $Λ_n = O(n^{1 + γ^\star})$, uniformly over Leja sequences. This improves the previously known best exponent $13/4 = 3.25$ to around $1.6304$ for such sets (including the benchmark case $K = [-1,1]$), hence substantially narrowing the gap with the near-linear growth observed numerically. Corresponding polynomial bounds for $τ$-Leja sequences are derived.
Source: arXiv (Cornell University)
Author: Pouchol, Camille
Institution: Mathématiques Appliquées à Paris 5
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7167210012