Geometric conditions for interpolation

Belna, C. L.; Obaid, Samih; Rung, D. C.

doi:10.1090/s0002-9939-1983-0699416-8

Proc. Amer. Math. Soc.

1983

DOI: 10.1090/s0002-9939-1983-0699416-8

|View full text |Cite

Geometric conditions for interpolation

C. L. Belna¹,

Samih Obaid²,

D. C. Rung³

Abstract: Abstract.Let (z") be a sequence lying in either the upper half-plane or the unit disc in the complex plane. If (z") 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.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

1985

2017

Publication Types

Select...

Article3

Relationship

Self Cite1

Independent2

Authors

Journals

Cited by 3 publications

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

The sharpeness of some cluster set results

Rung

Obaid

1986

International Journal of Mathematics and Mathematical Sciences

Self Cite

View full text Add to dashboard Cite

Abstract. We show that a recent cluster set theorem of Rung is sharp in a certain sense. This is accomplished through the construction of an interpolating sequence whose limit set Is c]osed, totally disconnected and porous. The results also generalize some of Dolzenko's cluster set theorems.Key words. Cluster sets, interpolating sequences, porous sets. 1980 Mathematics Subject Classification. Primary 30D40, Secondary 30E05. 1.INTERPOLATING SEQUENCES. We begin by considering a closed totally disconnected set P on the boundary 3a of the unit disc a in the complex plane. Thus =3a-P is the union of countably many disjoint arcs. Our first objective is to construct interpolating sequences on certain curves in the unit dlsc a whose llmlt points are all the points of P. In a speclal case Dolzenko [1] used this construction apparently not realizing that he was dealing wlth Interpolating sequences. We wish to define an approach to a point z a inside a reasonably nice subdomain of a. Let h(t) be a real-valued function defined for -1 < t < 1. We require that(1) h be continuous.(ii) h(t)=h(-t).(iii) h(O)=O, h(1)=1, h(t 1) _< h(t2), 0 _< t _< t 2 _< 1.(iv) h(t) _< t.Such an h is said to be a convex approach function. This function h determines a convex boundary domain (e,h) at z=e ie as follows (See Fig. I).}(e,h)

show abstract

The sharpeness of some cluster set results

Rung

Obaid

1986

International Journal of Mathematics and Mathematical Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

One-component inner functions

Cima

Mortini

2017

Complex Anal Synerg

View full text Add to dashboard Cite

Definition 1 An inner function u in H ∞ is said to be a one-component inner function if there is η ∈]0, 1[ such that the level set (also called sublevel set or filled level set) � u (η) := {z ∈ D : |u(z)| < η} is connected.One-component inner functions, the collection of which we denote by I c , were first studied by Cohn [10] in connection with embedding theorems and Carleson measures. It was shown that [for instance, [10], p. 355] arclength on {z ∈ D : |u(z)| = ε} is such a measure whenever is connected and η < ε < 1.A thorough study of the class I c was given by Aleksandrov [1] who showed the interesting result that u ∈ I c if and only if there is a constant C = C(u) such that for all a ∈ D Many operator-theoretic applications are given in [1][2][3]7]. In our paper here, we are interested in explicit examples, which are somewhat lacking in the literature. For example, if S is the atomic inner function, which is given by AbstractWe explicitly unveil several classes of inner functions u in H ∞ with the property that there is η ∈]0, 1[ such that the level set � u (η) := {z ∈ D : |u(z)| < η} is connected. These so-called one-component inner functions play an important role in operator theory.

show abstract

Interpolating sequences on curves

Obaid¹,

Rung²

1985

Rocky Mountain J. Math.

View full text Add to dashboard Cite

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Geometric conditions for interpolation

Cited by 3 publications

References 11 publications

The sharpeness of some cluster set results

The sharpeness of some cluster set results

One-component inner functions

Interpolating sequences on curves

Contact Info

Product

Resources

About