[ m at h . N A ] 1 6 M ay 2 01 5 New bounds on the Lebesgue constants of Leja sequences on the unit disc and on R - Leja sequences

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 U and 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 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 √ k when k has a sparse binary expansion. It also yields an improvement on the Lebesgue constants associated with R-Leja sequences.