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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.03623 |
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| _version_ | 1866917686291726336 |
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| author | Carando, Daniel Lassalle, Silvia Milne, Leandro |
| author_facet | Carando, Daniel Lassalle, Silvia Milne, Leandro |
| contents | Given a frequency $λ=(λ_n)$, we consider the Hardy spaces $ \mathcal{H}_p^λ$ of $λ$-Dirichlet series $ D = \sum_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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The fundamental functions of the canonical basis of Hardy spaces of Dirichlet series Carando, Daniel Lassalle, Silvia Milne, Leandro Functional Analysis 41A65, 43A17, 30B50, 46B15 Given a frequency $λ=(λ_n)$, we consider the Hardy spaces $ \mathcal{H}_p^λ$ of $λ$-Dirichlet series $ D = \sum_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. |
| title | The fundamental functions of the canonical basis of Hardy spaces of Dirichlet series |
| topic | Functional Analysis 41A65, 43A17, 30B50, 46B15 |
| url | https://arxiv.org/abs/2406.03623 |