Sets of sampling and interpolation in Bergman spaces

Schuster, Alexander

doi:10.1090/S0002-9939-97-03899-9

Sets of sampling and interpolation in Bergman spaces
HTML articles powered by AMS MathViewer

by Alexander P. Schuster PDF

Proc. Amer. Math. Soc. 125 (1997), 1717-1725 Request permission

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

Éric Amar, Suites d’interpolation pour les classes de Bergman de la boule et du polydisque de $\textbf {C}^{n}$, Canadian J. Math. 30 (1978), no. 4, 711–737 (French). MR 499309, DOI 10.4153/CJM-1978-062-6
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Peter Duren, Dmitry Khavinson, Harold S. Shapiro, and Carl Sundberg, Contractive zero-divisors in Bergman spaces, Pacific J. Math. 157 (1993), no. 1, 37–56. MR 1197044, DOI 10.2140/pjm.1993.157.37
Per Jan Håkan Hedenmalm, An invariant subspace of the Bergman space having the codimension two property, J. Reine Angew. Math. 443 (1993), 1–9. MR 1241125, DOI 10.1515/crll.1993.443.1
H. Hedenmalm, S. Richter and K. Seip, Interpolating sequences and invariant subspaces in the Bergman spaces (to appear).
Charles Horowitz, Zeros of functions in the Bergman spaces, Duke Math. J. 41 (1974), 693–710. MR 357747
Charles Horowitz, Factorization theorems for functions in the Bergman spaces, Duke Math. J. 44 (1977), no. 1, 201–213. MR 427650
A. Schuster, Ph.D. thesis, University of Michigan, Ann Arbor (to appear).
Kristian Seip, Regular sets of sampling and interpolation for weighted Bergman spaces, Proc. Amer. Math. Soc. 117 (1993), no. 1, 213–220. MR 1111222, DOI 10.1090/S0002-9939-1993-1111222-5
K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1994), 21-39.
K. Seip, On Korenblum’s density condition for the zero sequences of $A^{-\alpha }$ (to appear).
H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532. MR 133446, DOI 10.2307/2372892
Kam-fook Tse, Nontangential interpolating sequences and interpolation by normal functions, Proc. Amer. Math. Soc. 29 (1971), 351–354. MR 274777, DOI 10.1090/S0002-9939-1971-0274777-4
Dennis H. Wortman, Interpolating sequences on convex curves in the open unit disc, Proc. Amer. Math. Soc. 48 (1975), 157–164. MR 361092, DOI 10.1090/S0002-9939-1975-0361092-7
Ke He Zhu, Interpolating sequences for the Bergman space, Michigan Math. J. 41 (1994), no. 1, 73–86. MR 1260609, DOI 10.1307/mmj/1029004915
Ke He Zhu, Evaluation operators on the Bergman space, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 3, 513–523. MR 1317493, DOI 10.1017/S0305004100073333