Sampling sequences for Hardy spaces of the ball

Abstract:

We show that a sequence $a:=\{a_{k}\}_{k}$ in the unit ball of $\mathbb {C}^{n}$ is sampling for the Hardy spaces $H^{p}$, $0<p<\infty$, if and only if the admissible accumulation set of $a$ in the unit sphere has full measure. For $p=\infty$ the situation is quite different. While this condition is still sufficient, when $n>1$ (in contrast to the one dimensional situation) there exist sampling sequences for $H^{\infty }$ whose admissible accumulation set has measure 0. We also consider the sequence $a(\omega )$ obtained by applying to each $a_{k}$ a random rotation, and give a necessary and sufficient condition on $\{|a_{k}|\}_{k}$ so that, with probability one, $a(\omega )$ is of sampling for $H^{p}$, $p<\infty$.