Binomials whose dilations generate $H^2(\mathbb {D})$

Abstract:

This note is about the completeness of the function families \begin{equation*} \{z^n(\lambda -z^n)^N : n= 1,2,\dots \} \end{equation*} in the Hardy space $H^2_0(\mathbb {D})$, and some related questions. It is shown that for $|\lambda | > R(N)$ the family is complete in $H^2_0(\mathbb {D})$ (and often is a Riesz basis of $H^2_0$), whereas for $|\lambda | < r(N)$ it is not, where both radii $r(N)\leq R(N)$ tends to infinity and behave more or less as $N$ (as $N\to \infty$). Several results are also obtained for more general binomials $\{z^n(1-\frac {1}{\lambda } z^n)^{\nu } : n= 1,2,\dots \}$ where $|\lambda |\geq 1$ and $\nu \in \mathbb {C}$.