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arXiv · 2406.03623

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

Daniel Carando·Silvia Lassalle·Leandro Milne
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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.

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Daniel Carando, Silvia Lassalle, Leandro Milne. 2024-06-05. The fundamental functions of the canonical basis of Hardy spaces of Dirichlet series. https://arxiv.org/abs/2406.03623

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