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 ${\Lambda }_n$ of Leja sequences for every nonpolar compact set \(K\subset\mathbb C\): ${\mathrm{limsup}}_{n\to +\infty }{\Lambda }_n\sqrt{n^{-1}{W}_n(K)}\ge \frac{1}{\sqrt{2}}$, 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 ${\gamma }^{\star }\approx 0.6304$. 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 ${\Lambda }_n=O(n^{1+{\gamma }^{\star }})$, 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 $\tau$-Leja sequences are derived.