Lebesgue constants of Leja sequences: a universal lower bound and improved upper bounds on the real line.
Abstract
We prove a general lower bound for the Lebesgue constants Λn of Leja sequences for every nonpolar compact set K⊂ℂ: limsupn →+∞ Λn √(n−1 Wn(K))≥ 1/√2, where Wn(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⋆ ≈ 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 Λn = O(n1 + γ⋆), 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.
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BibTeX
@misc{pouchol2026lebesgue,
title = {{Lebesgue constants of Leja sequences: a universal lower bound and improved upper bounds on the real line.}},
author = {Camille Pouchol},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.01836},
eprint = {2607.01836},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05676129v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Lebesgue constants of Leja sequences: a universal lower bound and improved upper bounds on the real line.
AU - Camille Pouchol
PY - 2026
DA - 2026/09/05
PB - HAL
DO - 10.48550/arxiv.2607.01836
UR - https://hal.science/hal-05676129v2
AB - We prove a general lower bound for the Lebesgue constants Λ_n of Leja sequences for every nonpolar compact set K⊂ℂ: limsup_n →+∞ Λ_n √(n^−1 W_n(K))≥ 1/√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^⋆ ≈ 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^⋆), 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.
ER -