Sampling sets for Hardy spaces of the disk

Abstract

We propose two possible definitions for the notion of a sampling sequence (or set) for Hardy spaces of the disk. The first one is inspired by recent work of Bruna, Nicolau, and Øyma about interpolating sequences in the same spaces, and it yields sampling sets which do not depend on the value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and correspond to the result proved for bounded functions ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p=\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) by Brown, Shields and Zeller. The second notion, while formally closer to the one used for weighted Bergman spaces, leads to trivial situations only, but raises a possibly interesting problem.