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

N. K. Nikolski

Algebra i analiz, Tome 29 (2017) no. 6, pp. 159-177.

Résumé

This note is about the completeness of the function families $$ \{z^n(\lambda-z^n)^N\colon n=1,2,\dots\} $$ 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 $|\lambda|$ 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 $\{z^n(1-\frac1\lambda z^n)^\nu\colon n=1,2,\dots\}$ where $|\lambda|\geq1$ and $\nu\in\mathbb C$.

Détail

Keywords: Hardy spaces, completeness of dilations, Riesz basis, Hilbert multidisc, Bohr transform, binomial functions.

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