Projection constants for spaces of Dirichlet polynomials
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arXiv
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| Format: | Preprint |
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2023
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| author | Defant, Andreas Galicer, Daniel Mansilla, Martín Mastyło, Mieczysław Muro, Santiago |
| author_facet | Defant, Andreas Galicer, Daniel Mansilla, Martín Mastyło, Mieczysław Muro, Santiago |
| contents | Given a frequency sequence $ω=(ω_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(ω)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-ω_n s}, \, s \in \mathbb{C}$. The main aim is to prove asymptotically correct estimates for the projection constant $\boldsymbolλ\big(\mathcal{H}_\infty^{J}(ω) \big)$ of the finite dimensional Banach space $\mathcal{H}_\infty^{J}(ω)$ equipped with the norm $\|D\|= \sup_{\text{Re}\,s>0} |D(s)|$. Based on harmonic analysis on $ω$-Dirichlet groups, we prove the formula $ \boldsymbolλ\big(\mathcal{H}_\infty^{J}(ω) \big) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T \Big|\sum_{n \in J} e^{-iω_n t}\Big|\,dt\,, $ and apply it to various concrete frequencies $ω$ and index sets $J$. To see an example, combining with a recent deep result of Harper from probabilistic analytic number theory, we for the space $\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)$ of all ordinary Dirichlet polynomials $D(s) = \sum_{n \leq x} a_n n^{-s}$ of length $x$ show the asymptotically correct order $ \boldsymbolλ\big(\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)\big) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. $ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00231 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Projection constants for spaces of Dirichlet polynomials Defant, Andreas Galicer, Daniel Mansilla, Martín Mastyło, Mieczysław Muro, Santiago Functional Analysis Number Theory Primary: 30B50, 43A77, 46B06. Secondary: 46B07, 46B07, 46G25 Given a frequency sequence $ω=(ω_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(ω)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-ω_n s}, \, s \in \mathbb{C}$. The main aim is to prove asymptotically correct estimates for the projection constant $\boldsymbolλ\big(\mathcal{H}_\infty^{J}(ω) \big)$ of the finite dimensional Banach space $\mathcal{H}_\infty^{J}(ω)$ equipped with the norm $\|D\|= \sup_{\text{Re}\,s>0} |D(s)|$. Based on harmonic analysis on $ω$-Dirichlet groups, we prove the formula $ \boldsymbolλ\big(\mathcal{H}_\infty^{J}(ω) \big) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T \Big|\sum_{n \in J} e^{-iω_n t}\Big|\,dt\,, $ and apply it to various concrete frequencies $ω$ and index sets $J$. To see an example, combining with a recent deep result of Harper from probabilistic analytic number theory, we for the space $\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)$ of all ordinary Dirichlet polynomials $D(s) = \sum_{n \leq x} a_n n^{-s}$ of length $x$ show the asymptotically correct order $ \boldsymbolλ\big(\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)\big) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. $ |
| title | Projection constants for spaces of Dirichlet polynomials |
| topic | Functional Analysis Number Theory Primary: 30B50, 43A77, 46B06. Secondary: 46B07, 46B07, 46G25 |
| url | https://arxiv.org/abs/2302.00231 |