New Bounds on the Lebesgue Constants of Leja Sequences on the Unit Disc and on $\mathfrak {R}$-Leja Sequences

Abstract

In the papers [6, 7] we have established linear and quadratic bounds, in k, on the growth of the Lebesgue constants associated with the k-sections of Leja sequences on the unit disc $\mathcal{U}$ and $\mathfrak {R}$-Leja sequences obtained from the latter by projection into $[-1,1]$. In this paper, we improve these bounds and derive sub-linear and sub-quadratic bounds. The main novelty is the introduction of a “quadratic” Lebesgue function for Leja sequences on $\mathcal{U}$ which exploits perfectly the binary structure of such sequences and can be sharply bounded. This yields new bounds on the Lebesgue constants of such sequences, that are almost of order $\sqrt{k}$ when k has a sparse binary expansion. It also yields an improvement on the Lebesgue constants associated with $\mathfrak {R}$-Leja sequences.