On a Sufficient Condition for the Existence of Unconditional Bases of Reproducing Kernels in Hilbert Spaces of Entire Functions

Abstract

We consider a reproducing kernel radial Hilbert space of entire functions and prove a sufficient condition for the existence of unconditional bases of reproducing kernels in terms of norms of monomials. Let the system of monomials $\{\lambda^{n},\ n\in\mathbb{Z}_{+}\}$ is complete in a radial Hilbert space of entire functions $H$, and

$$u_{n}=\ln||\lambda^{n}||,\quad u_{+}^{\prime}(n)=u(n+1)-u(n),\quad n\in\mathbb{Z}_{+}.$$

If for some natural number $p$ the condition $\inf_{n}(u_{+}^{\prime}(n+p)-u_{+}^{\prime}(n))>0,$ holds, then $H$ possesses unconditional basis of reproducing kernels.