Random interpolating sequences in Dirichlet spaces

Nikolaos Chalmoukis (1) , Andreas Hartmann (2) , Karim Kellay (2) , Brett D Wick (3)

1 Dipartimento di Matematica
2 IMB - Institut de Mathématiques de Bordeaux
3 WUSTL - Washington University in Saint Louis

We discuss random interpolating sequences in weighted Dirichlet spaces $\mathcal{D}_\alpha$, $0\leq \alpha\leq 1$. Our results in particular imply that almost sure interpolating sequences for $\mathcal{D}_\alpha$ are exactly the almost sure separated sequences when $0\le \alpha<1/2$ (which covers the Hardy space $H^2=\mathcal{D}_0$), and they are exactly the almost sure zero sequences for $\mathcal{D}_\alpha$ when $1/2<\alpha<1$. We show that this last result remains valid in the classical Dirichlet space $\mathcal{D}=\mathcal{D}_1$ when one considers a weaker notion of interpolation, so-called simple interpolation. As a by-product we improve a sufficient condition by Rudowicz for random Carleson measures in Hardy spaces.

Mots clés: separation; Carleson measure; Interpolating sequences; random sequences

Domaines: Variables complexes [math.CV]

Nikolaos Chalmoukis, Andreas Hartmann, Karim Kellay, Brett D Wick. Random interpolating sequences in Dirichlet spaces. 2019. ⟨hal-02113238v1⟩