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Sampling sequences for Hardy spaces of the ball

Authors: Xavier Massaneda and Pascal J. Thomas
Journal: Proc. Amer. Math. Soc. 128 (2000), 837-843
MSC (1991): Primary 32A35, 32A30; Secondary 30B20, 30D50
Published electronically: July 28, 1999
MathSciNet review: 1646200
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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 $.

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Additional Information

Xavier Massaneda
Affiliation: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08071-Barcelona, Spain

Pascal J. Thomas
Affiliation: Laboratoire Emile Picard, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France

Received by editor(s): May 4, 1998
Published electronically: July 28, 1999
Additional Notes: Both authors were partially supported by a program of the Comunitat de Treball dels Pirineus. The second author was also supported by DGICYT grant PB95-0956-C02-01 and CIRIT grant GRQ94-2014.
Communicated by: Steven R. Bell
Article copyright: © Copyright 1999 American Mathematical Society

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