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.