Asymptotic greediness of the Haar system in the spaces $L_{p}[0,1]$, $1<p<\infty$

TL;DR

This paper provides a precise asymptotic estimate for the greedy constant of the Haar system in Lp spaces, showing it grows proportionally to p* as p approaches infinity, refining previous bounds.

Contribution

The paper establishes that the greedy constant of the Haar system in Lp spaces asymptotically behaves like p*, improving the understanding of greediness in these spaces.

Findings

The greedy constant C_g is asymptotically proportional to p* as p→∞.

The superdemocracy constant of the Haar system grows as p*.

The gap between known bounds for C_g is closed, showing C_g ≈ p*.

Abstract

Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant $C_g[\mathcal{H}^{(p)},L_p]$ of the (normalized) Haar system $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ for $1<p<\infty$. We will show that the superdemocracy constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ grows as $p^{\ast}=\max\{p,p/(p-1)\}$ as $p^*$ goes to $\infty$. Thus, since the unconditionality constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ is $p^*-1$, the well-known general estimates for the greedy constant of a greedy basis obtained from the intrinsic features of greediness (namely, democracy and unconditionality) yield that $p^{\ast}\lesssim C_g[\mathcal{H}^{(p)},L_p]\lesssim (p^{\ast})^{2}$. Going further, we develop techniques that allow us to close the gap between those two bounds, establishing that, in fact, $C_g[\mathcal{H}^{(p)},L_p]\approx p^{\ast}$.