Thin interpolating sequences in the disk

Mortini, Raymond

doi:10.1007/s00013-009-3057-x

Cited by 4 publications

(2 citation statements)

References 19 publications

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“…If the original sequence {z j } has the property that the corresponding Blaschke product C satisfies δ j (C) = |C j (z j )| → 1, then the function that does the interpolation may be chosen to have this property as well; that is, it can be chosen to be a so-called thin Blaschke product (this result was stated in [10] with a general plan of attack; details appear in [25]). The proof involves adapting the proof of J. P. Earl to this situation.…”

Section: Best Bounds and Best Functionsmentioning

confidence: 99%

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Thin Sequences

Gorkin,

Wick

2015

Preprint

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We look at thin interpolating sequences and the role they play in uniform algebras, Hardy spaces, and model spaces."One of the striking successes of the function algebra viewpoint has been in the study of the algebra H ∞ of all bounded holomorphic functions on the unit disk D. Not only are the results which have been obtained deep but the questions raised have also enriched the classical study of the boundary behavior of holomorphic functions." R. G. Douglas, Mathematical Reviews, (MR0428044 and MR428045)

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Section: Best Bounds and Best Functionsmentioning

confidence: 99%

“…In fact, the solution can be chosen to be a thin Blaschke product, as noted in [10]. (For the details of the proof, see [25]).…”

Section: Extending the Definition Of Thin To H P Spacesmentioning

confidence: 99%

Thin Sequences

Gorkin,

Wick

2015

Preprint

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Quasi-Universal Functions for Sequences of Composition Operators on H 2 Via Hoffman’s Theory

Mortini

2011

Integr. Equ. Oper. Theory

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Let φ be a non-elliptic automorphism of the unit disk D. Gallardo and Gorkin (respectively Gallardo, Gorkin and Suárez) showed that there exists an interpolating Blaschke product b (respectively a thin Blaschke product) such that the linear span of its orbit under the composition operator C φ is dense in H 2 (in other words that b is a cyclic vector for C φ ). Using Hoffman's theory, we extend their result from the automorphic case to the selfmap case and show that for any sequence (φn) of holomorphic selfmaps of D with |φn(0)| → 1 and lim sup n→∞ |φ n (0)| 1 − |φn(0)| 2 = 1, there exists a thin Blaschke product B with zero distribution as sparse as one wishes such that the linear span of the orbit {B • φn : n ∈ N} of B is dense in H 2 . We will dub such a function a quasi-universal function. We also describe the limit points in the topological space M (H ∞ ) D of these admissible sequences (φn). This approach will enable us to give shorter proofs of the Gallardo-Gorkin result as well as the result of Bayart, Gorkin, Grivaux and the author on the existence of B-universal functions for admissible sequences. Finally we discuss the problem whether interpolating Blaschke products can ever be B-universal for admissible sequences (φn).Mathematics Subject Classification (2010). Primary 47B33; Secondary 46J15, 47A16, 30J10.

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Thin sequences in the corona of H ∞

Stankov

Tzonev

2013

Open Mathematics

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In this paper we consider several conditions for sequences of points in M (H ∞ ) and establish relations between them. We show that every interpolating sequence for QA of nontrivial points in the corona M(H ∞ ) \ D of H ∞ is a thin sequence for H ∞ , which satisfies an additional topological condition. The discrete sequences in the Shilov boundary of H ∞ necessarily satisfy the same condition. MSC:30H05, 30H50, 46J15

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Thin interpolating sequences in the disk

Cited by 4 publications

References 19 publications

Thin Sequences

Thin Sequences

Quasi-Universal Functions for Sequences of Composition Operators on H 2 Via Hoffman’s Theory

Thin sequences in the corona of H ∞

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