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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2401.17520 |
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| _version_ | 1866911768304943104 |
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| author | Borichev, Alexander Fouchet, Karine Zarouf, Rachid |
| author_facet | Borichev, Alexander Fouchet, Karine Zarouf, Rachid |
| contents | Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell^{\infty}$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty$. We provide constructive examples which show that our estimates are sharp. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot\|T^{-1}\|\cdot\|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer's question on norms of inverses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17520 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Fourier coefficients of powers of a finite Blaschke product Borichev, Alexander Fouchet, Karine Zarouf, Rachid Complex Variables 30J10, 42A16, 41A60, 15A60 Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell^{\infty}$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty$. We provide constructive examples which show that our estimates are sharp. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot\|T^{-1}\|\cdot\|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer's question on norms of inverses. |
| title | On the Fourier coefficients of powers of a finite Blaschke product |
| topic | Complex Variables 30J10, 42A16, 41A60, 15A60 |
| url | https://arxiv.org/abs/2401.17520 |