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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
Language: Russian Download the full text  
For citation: Isaev, K. P. and Yulmukhametov, R. S. Unconditional Bases in Radial Hilbert Spaces, Vladikavkaz Math. J., 2020, vol. 22, no. 3, pp.85-99. DOI 10.46698/q8093-7554-9905-q
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