Schauder Basis with Finite Blaschke Products
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915767982751744 |
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| author | Fricain, Emmanuel Mashreghi, Javad Nasri, Mostafa Ostermann, Maëva |
| author_facet | Fricain, Emmanuel Mashreghi, Javad Nasri, Mostafa Ostermann, Maëva |
| contents | We construct a Schauder basis for the space $Hol(\mathbb D)$, the space of holomorphic functions on the closed unit disk, consisting entirely of finite Blaschke products. The expansion coefficients are given explicitly. Our result remains valid when $Hol(\mathbb D)$ is equipped with a broader class of norms satisfying natural structural conditions. These conditions are satisfied by norms of classical function spaces such as the Hardy spaces $H^p$ ($1\leq p\leq \infty$), the weighted Bergman spaces $A_α^p$ ($1\leq p\leq \infty$, $α>-1$), and BMOA. We also establish the optimality of this framework by proving that such a basis cannot exist in larger spaces, such as the Hardy space $H^p$ and the disc algebra $A(\mathbb D)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schauder Basis with Finite Blaschke Products Fricain, Emmanuel Mashreghi, Javad Nasri, Mostafa Ostermann, Maëva Complex Variables Functional Analysis We construct a Schauder basis for the space $Hol(\mathbb D)$, the space of holomorphic functions on the closed unit disk, consisting entirely of finite Blaschke products. The expansion coefficients are given explicitly. Our result remains valid when $Hol(\mathbb D)$ is equipped with a broader class of norms satisfying natural structural conditions. These conditions are satisfied by norms of classical function spaces such as the Hardy spaces $H^p$ ($1\leq p\leq \infty$), the weighted Bergman spaces $A_α^p$ ($1\leq p\leq \infty$, $α>-1$), and BMOA. We also establish the optimality of this framework by proving that such a basis cannot exist in larger spaces, such as the Hardy space $H^p$ and the disc algebra $A(\mathbb D)$. |
| title | Schauder Basis with Finite Blaschke Products |
| topic | Complex Variables Functional Analysis |
| url | https://arxiv.org/abs/2507.13121 |