Geometry of radial Hilbert spaces with unconditional bases of reproducing kernels

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
$$\begin{equation*} K(z)\asymp e^{2\widetilde u(\ln |z|)},\quad z\in \mathbb{C}. \end{equation*}$$