Abstract
The paper considers a parameterized family of Hilbert spaces of entire functions associated with a convex bounded set in the complex plane. For a convex bounded polygon \(D\) and a real parameter \(\beta\) , the space \(P_{\beta}(D)\) is introduced as the space of entire functions that are square integrable over \(D\) with weight \(\exp(-h_{D}(\arg)|z|-\beta\log(1+|z|))\) with respect to plane Lebesgue measure, where \(h_{D}(-\varphi)\) is the support function of \(D\) . It is proved that in each of these Hilbert spaces there exists an unconditional basis of reproducing kernels. In the case where \(D\) is a polygon with non-empty interior, this theorem implies the well-known result of B.Ya. Levin and Yu.I. Lyubarskii on the existence of unconditional bases of exponentials in the Smirnov space \(E_{2}(D)\) (for \(\beta=0\) ) and the result on the existence of unconditional bases of exponentials in the Bergman space \(B_{2}(D)\) (for \(\beta=-1/2\) ). If \(D\) is a segment, then this theorem in the dual formulation via Fourier–Laplace transforms means the existence Riesz bases of exponentials in Sobolev spaces \(H^{s}(D)\) .
