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Статьи
Binomials whose dilations generate $H^2(\mathbb D)$
N. K. Nikolskiab a Institute of Mathematics, University of Bordeaux, Bordeaux, France
b Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia
Аннотация:
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
$|\lambda|<r(N)$ 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 $\{z^n(1-\frac1\lambda z^n)^\nu\colon n=1,2,\dots\}$ where
$|\lambda|\geq1$ and
$\nu\in\mathbb C$.
Ключевые слова:
Hardy spaces, completeness of dilations, Riesz basis, Hilbert multidisc, Bohr transform, binomial functions.
MSC: 30H10 Поступила в редакцию: 09.08.2017
Язык публикации: английский
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