Unconditional bases in radial Hilbert spaces

K. P. Isaev   ; R. S. Yulmukhametov

Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 3, pp. 85-99

Résumé

We consider a Hilbert space of entire functions $H$ that satisfies the conditions: 1) $H$ is functional, that is the evaluation functionals $\delta _z: f\rightarrow f(z)$ are continuous for every $z\in \mathbb{C}$; 2) $H$ has the division property, that is, if $F\in H$, $F(z_0)=0$, then $F(z)(z-z_0)^{-1}\in H$; 3) $H$ is radial, that is, if $F\in H$ and $\varphi \in \mathbb R$, then the function $F(ze^{i\varphi })$ lies in $H$, and $\|F(ze^{i\varphi })\|= \|F\|$; 4) polynomials are complete in $H$ and $\|z^n\|\asymp e^{u(n)},$ $n\in \mathbb N\cup \{0\},$ where the sequence $u(n)$ satisfies the condition $u(n+1)+u(n-1)-2u(n)\succ n^\delta ,$ $n\in \mathbb N,$ for some $\delta >0$. It follows from condition 1) that every functional $\delta _z$ is generated by an element $k_z(\lambda )\in H$ in the sense of $\delta _z(f)=(f(\lambda ),k_z(\lambda )).$ The function $k(\lambda, z)=k_z(\lambda )$ is called the reproducing kernel of the space $H$. A basis $\{ e_k,\ k=1,2,\ldots\}$ in Hilbert space $H$ is called a unconditional basis if there exist numbers $c,C > 0$ such that for any element $x=\sum \nolimits _{k=1}^{\infty } x_ke_k\in H$ the relation $$ c\sum _{k=1}^\infty |c_k|^2\|e_k\|^2\le \left \|x \right \|^2\le C\sum _{k=1}^\infty |c_k|^2\|e_k\|^2 $$ holds true. The article describes a method for constructing unconditional bases of reproducing kernels in such spaces. This problem goes back to two closely related classical problems: representation of functions by series of exponentials and interpolation by entire functions.

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