Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-99-05212-0
Published electronically: July 28, 1999
MathSciNet review: 1646200
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $.


References [Enhancements On Off] (What's this?)

  • [Br-Ni-Oy] Bruna J. - Nicolau A. - Øyma K., A note on interpolation in the Hardy spaces in the disc, Proc. Amer. Math. Soc. 124 (1996), 1197-1204. MR 96g:30066
  • [Br-Sh-Ze] Brown L., Shields A., Zeller K., On absolutely convergent exponential sums, Trans. Amer. Math. Soc. 96 (1960), 162-183. MR 26:332
  • [Bo] Bomash G., A Blaschke-type product and random zero sets for Bergman spaces, Ark. Mat. 30 (1992), 45-60. MR 93g:30047
  • [Co] Cochran W. G., Random Blaschke products, Trans. Amer. Math. Soc. 332 (1990), 731-755. MR 91c:30061
  • [It] Itô K., Introduction to probability theory, Cambridge University Press, 1978. MR 86k:60001
  • [Ma] Massaneda X., Random sequences with prescribed radii in the unit ball, Complex Variables 31 (1996), 193-211. MR 98e:32006
  • [Rd] Rudowicz R., Random interpolating sequences with probability one, Bull. London Math. Soc. 26 (1994), 160-164. MR 95k:30073
  • [Ru1] Rudin W., Function theory in the unit ball of $\mathbb{C}^{n}$, Springer Verlag, Berlin, 1980. MR 82i:32002
  • [Ru2] Rudin W., New constructions of functions holomorphic in the unit ball of $\mathbb{C}^{n}$, CBMS Regional Conf. Ser. in Math. 63 AMS, Providence, 1986. MR 87f:32013
  • [Th] Thomas P.J., Sampling sets for Hardy spaces of the disk, Proc. Amer. Math. Soc. 126 (1998), 2927-2932. CMP 98:16

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 32A35, 32A30, 30B20, 30D50

Retrieve articles in all journals with MSC (1991): 32A35, 32A30, 30B20, 30D50


Additional Information

Xavier Massaneda
Affiliation: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08071-Barcelona, Spain
Email: xavier@cerber.mat.ub.es

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

DOI: https://doi.org/10.1090/S0002-9939-99-05212-0
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

American Mathematical Society