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Complete interpolating sequences for Fock type spaces

Karim Kellay, Youssef Omari

Abstract

We obtain a characterization of complete interpolating sequences in a class of Fock-type spaces with radial weights for which such sequences exist. Our criterion is formulated in terms of logarithmic separation and controlled perturbations of a reference sequence satisfying an Avdonin-type condition. This provides a geometric description of complete interpolating sequences and extends previous results of Borichev--Lyubarskii and Baranov--Belov--Borichev on Riesz bases of reproducing kernels in Fock-type spaces. It also yields explicit density criteria for sampling and interpolating sequences.

Complete interpolating sequences for Fock type spaces

Abstract

We obtain a characterization of complete interpolating sequences in a class of Fock-type spaces with radial weights for which such sequences exist. Our criterion is formulated in terms of logarithmic separation and controlled perturbations of a reference sequence satisfying an Avdonin-type condition. This provides a geometric description of complete interpolating sequences and extends previous results of Borichev--Lyubarskii and Baranov--Belov--Borichev on Riesz bases of reproducing kernels in Fock-type spaces. It also yields explicit density criteria for sampling and interpolating sequences.
Paper Structure (14 sections, 17 theorems, 225 equations)

This paper contains 14 sections, 17 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Reference sequence and complete interpolation sequences for ${\mathcal{F}}^p_\varphi$
  3. Density results
  4. Preliminary Results
  5. First evaluation estimates.
  6. Canonical products estimates
  7. Reference complete interpolating sequence for ${\mathcal{F}}^p_\varphi$.
  8. Asymptotic evaluation estimates
  9. Separated sequences
  10. Proof of Theorem \ref{['thm1']}
  11. Sufficient conditions
  12. Necessary conditions
  13. Proof of Theorem \ref{['inftyThm']}.
  14. Proof of Theorem \ref{['densitythm']}

Key Result

Theorem 1.1

Let $\Gamma=\{\gamma_n\}_{n\ge0}$ be a sequence of $\mathbb{C}$ such that $|\gamma_n|\le |\gamma_{n+1}|$ and write $\gamma_n=e^{y_n}e^{\delta_n}e^{i\theta_n}$, for some real sequences $(\delta_n)$ and $(\theta_n)$. Then $\Gamma$ is a complete interpolating sequence for ${\mathcal{F}}^p_\varphi$, $0<

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 18 more