New Bounds on the Lebesgue Constants of Leja Sequences on the Unit Disc and on R-Leja Sequences

摘要

In the papers 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~(1/2) when k has a sparse binary expansion. It also yields an improvement on the Lebesgue constants associated with R-Leja sequences.