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arXiv · 1911.11001

Riesz bases of reproducing kernels in small Fock spaces

K. Kellay·Youssef Omari
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Abstract

We give a complete characterization of Riesz bases of normalized reproducing kernels in the small Fock spaces $\mathcal{F}^2_φ$, the spaces of entire functions $f$ such that $f\mathrm{e}^{-φ} \in L^{2}(\mathbb{C})$, where $φ(z)= (\log^+|z|)^{β+1}$, $0< β\leq 1$.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-Dumont-Hartman-Kellay in the case when $β=1$. The present paper answers a question in Baranov et al. by extending their results for all parameters $β\in (0,1)$. 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.

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K. Kellay, Youssef Omari. 2019-11-25. Riesz bases of reproducing kernels in small Fock spaces. https://arxiv.org/abs/1911.11001

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