Schauder Basis with Finite Blaschke Products

TL;DR

This paper constructs a Schauder basis for holomorphic functions on the unit disk using finite Blaschke products, valid across various classical function spaces, and proves the limitations of such bases in larger spaces.

Contribution

It introduces a Schauder basis composed of finite Blaschke products for $Hol( extbf{D})$ and related spaces, extending the understanding of bases in complex function spaces.

Findings

Basis exists for $Hol( extbf{D})$ and classical spaces like Hardy and Bergman spaces.

Such a basis cannot exist in larger spaces like $H^p$ and $A( extbf{D})$.

The basis coefficients are explicitly given.

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 $A(\mathbb D)$.