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arXiv · 2507.13121

Schauder Basis with Finite Blaschke Products

Emmanuel Fricain·Javad Mashreghi·Mostafa Nasri·Maëva Ostermann
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Abstract

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 $A_\alpha^p$ ($1\leq p\leq \infty$, $\alpha>-1$), 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 $A(\mathbb D)$.

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Emmanuel Fricain, Javad Mashreghi, Mostafa Nasri, Maëva Ostermann. 2025-07-17. Schauder Basis with Finite Blaschke Products. https://arxiv.org/abs/2507.13121

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