Sampling sets for Hardy spaces of the disk

Author:
Pascal J. Thomas

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2927-2932

MSC (1991):
Primary 30E10, 30D55, 30C15

DOI:
https://doi.org/10.1090/S0002-9939-98-04411-6

MathSciNet review:
1452830

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Abstract | References | Similar Articles | Additional Information

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 and correspond to the result proved for bounded functions () 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.

**[Br-Ni-Øy]**Bruna J., Nicolau A., Øyma K.,*A note on interpolation in the Hardy spaces of the unit 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****[Du]**Duren P.,*Theory of Spaces*, Academic Press, New York, 1970. MR**42:3552****[Ga]**Garnett J.,*Bounded analytic functions*, Academic Press, New York, 1981. MR**83g:30037****[La]**Landau H.J.,*Necessary density conditions for sampling and interpolation of certain entire functions*, Acta Math.**117**(1967), 37-52. MR**36:5604****[Lu]**Luecking D.,*Dominating measures for spaces of analytic functions*, Ill. J. Math.**32**(1988), 23-39. MR**89e:46060****[Ly-Se]**Lyubarskii Yu., Seip K.,*A uniqueness theorem for bounded analytic functions*, Bull. London Math. Soc.**29**(1997), 49-52. MR**97m:30037****[Se1]**Seip K.,*Beurling type density theorems in the unit disk*, Invent. Math.**113**(1993), 21-39. MR**94g:30033****[Se2]**Seip K.,*Regular sets of sampling and interpolation for weighted Bergman spaces*, Proc. Amer. Math. Soc.**117**(1993), 213-220. MR**93c:30051****[Sh-Sh]**Shapiro H. S., Shields A.,*On some interpolation problems for analytic functions*, Amer. J. Math.**83**(1961), 513-532. MR**24:A3280****[St]**Stein E. M.,*Singular integrals and differentiability properties of functions*, Princeton University Press, Princeton, 1970. MR**44:7280**

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Additional Information

**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/S0002-9939-98-04411-6

Received by editor(s):
October 14, 1996

Received by editor(s) in revised form:
February 27, 1997

Communicated by:
Theodore W. Gamelin

Article copyright:
© Copyright 1998
American Mathematical Society