Riesz bases of reproducing kernels in small Fock spaces

Karim Kellay (1) , Youssef Omari

1 IMB - Institut de Mathématiques de Bordeaux

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 $\varphi(z)= (\log^+|z|)^{\beta+1}$, $0< \beta \leq 1$. 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-Dumont-Hartman-Kellay 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.

Domaines: Variables complexes [math.CV]; Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]

Karim Kellay, Youssef Omari. Riesz bases of reproducing kernels in small Fock spaces. Journal of Fourier Analysis and Applications, 2020, 26 (1). ⟨hal-01918516v2⟩