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

2024/06/05 by Daniel Carando, Carando, Daniel, Silvia Lassalle +3
Mathematics · #30B50 #41A65 #43A17 #46B15 #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2406.03623

openalex publication_date 2024/06/05 · openalex created_date 2024/06/08 · openalex updated_date 2026/08/01

Abstract

Given a frequency λ=(λ_n), we consider the Hardy spaces H_p^λ of λ-Dirichlet series D = ∑_n a_n e^{-λ_n s} and study the asymptotic behavior of the upper and lower democracy functions of its canonical basis \mathcal B=\e^-λ_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 λ lead to particular behaviors of the corresponding fundamental functions. Finally, we apply our results to analyze greedy-type properties of \mathcal B=\e^-λ_ns\ for some particular λ's.

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