Schauder Basis with Finite Blaschke Products

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

Theorems & Definitions (28)

• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof
• Remark 3.4
• Proposition 3.5
• proof