Abstract
We give a complete characterization of Riesz bases of normalized reproducing kernels in the small Fock spaces $${{\mathcal {F}}}^2_{\varphi }$$, the spaces of entire functions f such that $$f\mathrm {e}^{-\varphi } \in L^{2}({\mathbb {C}})$$, where $$\varphi (z)= (\log ^+|z|)^{\beta +1}$$, $$0< \beta \le 1$$. The first results in this direction are due to Borichev–Lyubarskii who showed that $$\varphi $$ 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 $$\beta \in (0,1)$$. 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.
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