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

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that a sequence in the unit ball of is sampling for the Hardy spaces , , if and only if the admissible accumulation set of in the unit sphere has full measure. For the situation is quite different. While this condition is still sufficient, when (in contrast to the one dimensional situation) there exist sampling sequences for whose admissible accumulation set has measure 0. We also consider the sequence obtained by applying to each a random rotation, and give a necessary and sufficient condition on so that, with probability one, is of sampling for , .

**[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*, Springer Verlag, Berlin, 1980. MR**82i:32002****[Ru2]**Rudin W.,*New constructions of functions holomorphic in the unit ball of*, 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**

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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