The fundamental functions of the canonical basis of Hardy spaces of Dirichlet series

TL;DR

This paper investigates the asymptotic behavior of democracy functions of the canonical basis in Hardy spaces of Dirichlet series, revealing bounds, examples, and properties related to the frequency sequence and greedy algorithms.

Contribution

It provides sharp bounds and characterizations of democracy functions for Hardy spaces of Dirichlet series, including examples of intermediate behaviors and applications to greedy algorithms.

Findings

Sharp asymptotic bounds for democracy functions in the ordinary case.

Examples demonstrating all intermediate behaviors for p>2.

Analysis of how frequency properties influence basis behavior.

Abstract

Given a frequency $\lambda=(\lambda_n)$, we consider the Hardy spaces $ \mathcal{H}_p^\lambda$ of $\lambda$-Dirichlet series $ D = \sum_n a_n e^{-\lambda_n s}$ and study the asymptotic behavior of the upper and lower democracy functions of its canonical basis $\mathcal B=\{e^{-\lambda_ns}\}$. For the ordinary case, $\mathcal B=\{n^{-s}\}$, we give the correct asymptotic behavior of all such functions, while in the general case we give sharp lower and upper bounds for all possible behaviors. Moreover, for $p>2$ we present examples showing that any intermediate behavior (between the extreme bounds) can occur. We also study how different properties of the frequency $\lambda$ lead to particular behaviors of the corresponding fundamental functions. Finally, we apply our results to analyze greedy-type properties of $\mathcal B=\{e^{-\lambda_ns}\}$ for some particular $\lambda$'s.