New bounds on the Lebesgue constants of Leja sequences on the unit disc and their projections $\Re$-Leja sequences

TL;DR

This paper improves bounds on the Lebesgue constants for Leja sequences on the unit disc and their projections, introducing a quadratic Lebesgue function that leverages binary structure for sharper estimates, especially for sparse binary expansions.

Contribution

It introduces a quadratic Lebesgue function for Leja sequences, leading to improved sub-linear and sub-quadratic bounds on Lebesgue constants, enhancing understanding of their growth behavior.

Findings

Bounds on Lebesgue constants are nearly of order √k for sparse binary k.

New quadratic Lebesgue function exploits binary structure of sequences.

Improved bounds on Lebesgue constants for Re-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 $\Re$-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 $\Re$-Leja sequences.