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 φ 2 , the spaces of entire functions f such that f e - φ ∈ L 2 ( C ) , where φ ( z ) = ( log + | z | ) β + 1 , 0 < β ≤ 1 . The first results in this direction are due to Borichev–Lyubarskii who showed that φ with β = 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 β = 1 . The present paper answers a question in Baranov et al. by extending their results for all parameters β ∈ ( 0 , 1 ) . Our results are analogous to those obtained for the case β = 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.