Geometry of radial Hilbert spaces with unconditional bases of reproducing kernels

K. P. Isaev ; R. S. Yulmukhametov

Ufa mathematical journal, Tome 12 (2020) no. 4, pp. 55-63

Résumé

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*}

Détail

Keywords: Hilbert spaces, entire functions, unconditional bases, reproducing kernels.