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)



Sets of sampling and interpolation
in Bergman spaces

Author: Alexander P. Schuster
Journal: Proc. Amer. Math. Soc. 125 (1997), 1717-1725
MSC (1991): Primary 30H05, 46E15
MathSciNet review: 1396996
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Properties of the unions of sampling and interpolation sets for Bergman spaces are discussed in conjunction with the examples given by Seip (1993). Their relationship to the classical interpolation sequences is explored. In addition, the role played by canonical divisors in the study of these sets is examined and an example of a sampling set is constructed in the disk.

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

  • 1. E. Amar, Suites d'interpolation pour les classes de Bergman de la boule et du polydisque de $\mathbb {C}^{n}$, Canadian J. Math. 30 (1978), 711-737. MR 58:17208
  • 2. P.L. Duren, Theory of $H^{p}$ Spaces, Academic Press, New York, 1970. MR 42:3552
  • 3. P. Duren, D. Khavinson, H.S. Shapiro and C. Sundberg, Contractive zero-divisors in Bergman spaces, Pacific J. Math. 157 (1993), 37-56. MR 94c:30048
  • 4. H. Hedenmalm, An invariant subspace of the Bergman space having the codimension two property, J. Reine Angew. Math. 443 (1991), 1-9. MR 94k:30092
  • 5. H. Hedenmalm, S. Richter and K. Seip, Interpolating sequences and invariant subspaces in the Bergman spaces (to appear).
  • 6. C. Horowitz, Zeros of functions in the Bergman spaces, Duke Math. J. 41 (1974), 693-710. MR 50:10215
  • 7. C. Horowitz, Factorization theorems for functions in the Bergman spaces, Duke Math. J. 44 (1977), 201-213. MR 55:681
  • 8. A. Schuster, Ph.D. thesis, University of Michigan, Ann Arbor (to appear).
  • 9. K. Seip, Regular sets of sampling and interpolation for weighted Bergman spaces, Proc. Amer. Math. Soc. 117 (1993), 213-220. MR 93c:30051
  • 10. K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1994), 21-39.
  • 11. K. Seip, On Korenblum's density condition for the zero sequences of $A^{-\alpha }$ (to appear).
  • 12. H.S. Shapiro and A.L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513-532. MR 24:A3280
  • 13. K-F. Tse, Nontangential interpolating sequences and interpolation by normal functions, Proc. Amer. Math. Soc. 29 (1971), 351-354. MR 43:537
  • 14. D.H. Wortman, Interpolating sequences on convex curves in the open unit disc, Proc. Amer. Math. Soc. 48 (1975), 157-164. MR 50:13538
  • 15. K. Zhu, Interpolating sequences for the Bergman space, Michigan Math. J. 41 (1994), 73-86. MR 94m:46045
  • 16. K. Zhu, Evaluation operators on the Bergman space, Math. Proc. Camb. Phil. Soc. 117 (1995), 513-523. MR 95k:30077

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30H05, 46E15

Retrieve articles in all journals with MSC (1991): 30H05, 46E15

Additional Information

Alexander P. Schuster
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Keywords: Bergman space, sampling, interpolation, zero set, canonical divisor
Received by editor(s): December 7, 1995
Additional Notes: The content of this paper forms a part of the author’s doctoral dissertation at the University of Michigan, written under the direction of Professor Peter Duren, whose help is greatly appreciated.
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society