Unconditional bases in radial Hilbert spaces

We consider a Hilbert space of entire functions H that satisfies the conditions: 1) H is functional, that is the evaluation functionals δz:f→f(z) are continuous for every z∈C; 2) H has the division property, that is, if F∈H, F(z0)=0, then F(z)(z-z0)-1∈H; 3) H is radial, that is, if F∈H and φ∈R, then the function F(zeiφ) lies in H, and ∥F(zeiφ)∥=∥F∥; 4) polynomials are complete in H and ∥zn∥≍eu(n), n∈N∪{0}, where the sequence u(n) satisfies the condition u(n+1)+u(n-1)-2u(n)≻nδ, n∈N, for some δ>0. It follows from condition 1) that every functional δz is generated by an element kz(λ)∈H in the sense of δz(f)=(f(λ),kz(λ)). The function k(λ,z)=kz(λ) is called the reproducing kernel of the space H. A basis {ek, k=1,2,…} in Hilbert space H is called a unconditional basis if there exist numbers c,C>0 such that for any element x=∑∞k=1xkek∈H the relation c∑∞k=1|ck|2∥ek∥2≤∥x∥2≤C∑∞k=1|ck|2∥ek∥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.