Article
Ufa Mathematical Journal
Volume 12, Number 4, pp. 55-63
Geometry of radial Hilbert spaces with unconditional bases of reproducing kernels
Isaev K.P., Yulmukhametov R.S.
DOI:10.13108/2020-12-4-55
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We study the geometry of abstract radial functional Hilbert spaces
stable with respect to dividing and possessing an unconditional basis of reproducing kernels. We obtain a simple necessary condition ensuring the existence of such bases in terms of the sequence $\| z^n\|$, $n\in \mathbb{N}\cup \{ 0\}$. We also obtain a sufficient condition for the norm and the Bergman function of the space to be recovered by a sequence of the norms of monomials. Two main statements we prove are as follows. Let $ H $ be a radial functional Hilbert space of entire functions stable with respect to dividing and let the system of monomials $\{z^n\}$, $n\in \mathbb{N}\cup \{ 0\}$, be complete in this space.
1. If the space $H$ possesses an unconditional basis of reproducing kernels, then
\begin{equation*}
\|z^n\| \asymp e^{u(n)},\quad n\in \mathbb{N}\cup \{0\},
\end{equation*}
where the sequence $u(n)$ is convex, that is
\begin{equation*}
u(n+1)+u(n-1)-2u(n)\ge 0,\quad n\in \mathbb{N}.
\end{equation*}
2. Let $u_{n,k}=u(n)-u(k)-(u(n)-u(n-1))(n-k)$. If $\mathcal U$ is the matrix with entries $e^{2u_{n,k}}$, $n,k\in \mathbb{N}\cup \{ 0\}$, and
\begin{equation*}
\left \| \mathcal U\right \| :=\sup_n\left (\sum\limits_ke^ {2u_{n,k}}\right )^{\frac 12}<\infty ,
\end{equation*}
then
2.1. the space $H$ as a Banach space is isomorphic to the space of entire functions with the norm
\begin{equation*}
\|F\|^2=\frac 1 {2\pi }\int\limits_0^\infty \int\limits_0^{2\pi }|F(re^{i\varphi }) |^2e^{-2\widetilde u(\ln r)}d\varphi d \widetilde u_+'(\ln r),
\end{equation*}
where $\widetilde u$ is the Young conjugate of the piecewise-linear function $u(t)$;
2.2. the Bergman function of the space $H$ satisfies the condition
\begin{equation*}
K(z)\asymp e^{2\widetilde u(\ln |z|)},\quad z\in \mathbb{C}.
\end{equation*}