Sampling sequences for Hardy spaces of the ball

Abstract

We show that a sequence a:= {ak}k in the unit ball of Cn is sampling for the Hardy spaces Hp, 0 < p < ∞ , if and only if the admissible accumulation set of a in the unit sphere has full measure. For p = ∞ 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∞ whose admissible accumulation set has measure 0. We also consider the sequence a(ω ) obtained by applying to each ak a random rotation, and give a necessary and sufficient condition on {|ak|}k so that, with probability one, a(ω ) is of sampling for Hp, p < ∞ .