GEOMETRY OF RADIAL HILBERT SPACES WITH UNCONDITIONAL BASES OF REPRODUCING KERNELS
We study the geometry of abstract radial functional Hilbert spaces stable with respect to the dividing and possessing an unconditional basis of reproducing kernels. We obtain a simple necessary condition ensuring the existence of such bases in terms of the sequence ‖𝑧𝑛‖, ∈ N∪ {0}. We also we obtain a sufficient condition for the norm and the Bergman function of the space to be recovered by a sequence of the norms of monomial. Two main statements we prove are as follows. Let be a radial functional Hilbert space of entire functions stable with respect to the dividing and let the system of monomials {𝑧𝑛}, ∈ N ∪ {0}, be complete in this space. 1. If the space possesses an unconditional basis of reproducing kernels, then ‖𝑧𝑛‖ ≍ 𝑒𝑢(𝑛), ∈ N ∪ {0}, where the sequence 𝑢(𝑛) is convex, that is 𝑢(𝑛 + 1) + 𝑢(𝑛 - 1) - 2𝑢(𝑛) > 0, ∈ N. 2. Let 𝑢𝑛,𝑘 = 𝑢(𝑛)-𝑢(𝑘)-(𝑢(𝑛)-𝑢(𝑛-1))(𝑛-𝑘). If is the matrix with entries 𝑒2𝑢𝑛,𝑘 , 𝑛, ∈ N ∪ {0}, and ‖𝒰‖ := sup (︃ Σ︁ 𝑒2𝑢𝑛,𝑘 )︃1 2 < ∞, then 2.1. the space as a Banach space is isomorphic to the space of entire functions with the norm ‖𝐹‖2 = 1 2𝜋 ∫︁ ∞ 0 ∫︁ 2𝜋 0 |𝐹(𝑟𝑒𝑖𝜙)|2𝑒-2̃︀𝑢(ln 𝑟)𝑑𝜙𝑑̃︀𝑢′ +(ln 𝑟), where ̃︀𝑢 is the Young conjugate of the piecewise-linear function 𝑢(𝑡); 2.2. the Bergman function of the space satisfies the condition 𝐾(𝑧) ≍ 𝑒2̃︀𝑢(ln |𝑧|), ∈ C.