Random Interpolating Sequences in Dirichlet Spaces

Abstract

Abstract We discuss random interpolating sequences in weighted Dirichlet spaces ${{\mathcal{D}}}_\alpha $, $0\leq \alpha \leq 1$, when the radii of the sequence points are fixed a priori and the arguments are uniformly distributed. Although conditions for deterministic interpolation in these spaces depend on capacities, which are very hard to estimate in general, we show that random interpolation is driven by surprisingly simple distribution conditions. As a consequence, we obtain a breakpoint at $\alpha =1/2$ in the behavior of these random interpolating sequences showing more precisely that almost sure interpolating sequences for ${{\mathcal{D}}}_\alpha $ are exactly the almost sure separated sequences when $0\le \alpha <1/2$ (which includes the Hardy space $H^2={{\mathcal{D}}}_0$), and they are exactly the almost sure zero sequences for ${{\mathcal{D}}}_\alpha $ when $1/2 \leq \alpha \le 1$ (which includes the classical Dirichlet space ${{\mathcal{D}}}={{\mathcal{D}}}_1$).