Riesz Bases of Reproducing Kernels in Small Fock Spaces

We give a complete characterization of Riesz bases of normalized reproducing kernels in the small Fock spaces F-2., the spaces of entire functions f such that f e(-phi) is an element of L-2(C), where phi(z) = (log(+) vertical bar z vertical bar)(beta+1), 0 < beta <= 1. The first results in this direction are due to Borichev-Lyubarskii who showed that. with beta = 1 is the largest weight for which the corresponding Fock space admits Riesz bases of reproducing kernels. Later, such bases were characterized by Baranov et al. in the case when beta = 1. The present paper answers a question in Baranov et al. by extending their results for all parameters beta. (0, 1). Our results are analogous to those obtained for the case beta = 1 and those proved for Riesz bases of complex exponentials for the Paley-Wiener spaces. We also obtain a description of complete interpolating sequences in small Fock spaces with corresponding uniform norm.