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DOI: 10.46698/q8093-7554-9905-q

# Unconditional Bases in Radial Hilbert Spaces

Isaev, K. P. , Yulmukhametov R. S.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 3.
Abstract:
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.
Keywords: Hilbert spaces, entire functions, unconditional bases, reproducing kernels