Geometric conditions for interpolation

Belna, C. L.; Obaid, S. A.; Rung, D. C.

doi:10.1090/S0002-9939-1983-0699416-8

Geometric conditions for interpolation
HTML articles powered by AMS MathViewer

by C. L. Belna, S. A. Obaid and D. C. Rung PDF

Proc. Amer. Math. Soc. 88 (1983), 469-475 Request permission

Abstract:

Let $\left \{ {{z_n}} \right \}$ be a sequence lying in either the upper half-plane or the unit disc in the complex plane. If $\left \{ {{z_n}} \right \}$ is a separated sequence we give a simple geometric condition that implies the sequence is an interpolating sequence for the algebra of bounded holomorphic functions. This result contains most of the known results of this type.

References

Lennart Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930. MR 117349, DOI 10.2307/2372840
John Garnett, Interpolating sequences for bounded harmonic functions, Indiana Univ. Math. J. 21 (1971/72), 187–192. MR 284589, DOI 10.1512/iumj.1971.21.21016
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Eric Alan Gerber and Max L. Weiss, Interpolation, trivial and nontrivial homomorphisms in $H^{\infty }$, J. Math. Soc. Japan 34 (1982), no. 1, 173–185. MR 639811, DOI 10.2969/jmsj/03410173
W. Hayman, Interpolation by bounded functions, Ann. Inst. Fourier (Grenoble) 8 (1958), 277–290 (English, with French summary). MR 117348
Paul Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
D. J. Newman, Interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 92 (1959), 501–507. MR 117350, DOI 10.1090/S0002-9947-1959-0117350-X
Max L. Weiss, Some $H^{\infty }$-interpolating sequences and the behavior of certain of their Blaschke products, Trans. Amer. Math. Soc. 209 (1975), 211–223. MR 372219, DOI 10.1090/S0002-9947-1975-0372219-X
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