Fundamental functions of almost greedy bases of $L_p$ for $1<p< \infty$

TL;DR

This paper characterizes the growth of fundamental functions of almost greedy bases in $L_p$ spaces, showing they are either like $m^{1/p}$ or $m^{1/2}$, clarifying their structural behavior.

Contribution

It establishes a precise classification of the fundamental functions for almost greedy bases in $L_p$ spaces, revealing their possible growth patterns.

Findings

Fundamental functions grow as $m^{1/p}$ or $m^{1/2}$.

Provides a complete characterization of almost greedy bases in $L_p$.

Clarifies the structure of bases in Banach spaces.

Abstract

We prove that the fundamental function of any almost greedy basis of $L_p$, $1<p<\infty$, grows as either $(m^{1/p})_{m=1}^\infty$ or $(m^{1/2})_{m=1}^\infty$.