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Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants

Authors: Hartmann, Andreas; Massaneda, Xavier; Nicolau, Artur; Thomas, Pascal J.;

Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants

Abstract

Let \(X\) be a class of holomorphic functions on the unit disc \({\mathbb D}\). A discrete sequence \(\Lambda(\subset{\mathbb D})\) is called interpolating for \(X\) if \(X| _{\Lambda}\) is an ideal in \(\ell^\infty\). The set of all interpolating sequences for \(X\) is denoted by \(\text{Int }X\). For any algebra \(X\) containing the constants the Blaschke condition \(\sum_\Lambda(1-| \lambda| )0\). The authors consider a free interpolation problem in the Nevanlinna class \(N\) and the Smirnov class \(N^+\). Let \(\varphi_\Lambda(z)=\text{log }| B_\lambda(\lambda)| ^{-1}\) if \(z=\lambda\in\Lambda\), and \(\varphi_\Lambda(z)=0\) otherwise. It is shown that \(\Lambda\in\text{Int } N\) if and only if the function \(\varphi_\Lambda\) admits a harmonic majorant. A positive harmonic majorant \(h(z)\) is called \textit{quasi-bounded} if it is a Poisson integral of an absolutely continuous positive measure. A sequence \(\Lambda\) is shown to be in \(\text{Int } N^+\) if and only if the function \(\varphi_\Lambda\) admits a quasi-bounded harmonic majorant. The class of functions which admit harmonic (quasi-bounded) majorant is characterized in dual terms. Simple geometric necessary conditions for \(\Lambda\) to be in \(\text{Int } N\) (\(\text{Int } N^+\)) are obtained: \(\Lambda\in\text{Int } N\Longrightarrow\lim_{| \lambda| \to 1}(1-| \lambda| )\log| B_\lambda(\lambda)| ^{-1}<\infty\), \(\Lambda\in\text{Int } N^+\Longrightarrow\lim_{| \lambda| \to 1}(1-| \lambda| )\log| B_\lambda(\lambda)| ^{-1}=0\). When \(\Lambda\) lies in a finite union of Stolz angles these implications are shown to be equivalences and the former one gives partially a result of \textit{A.G. Naftalevič} [Vilniaus Valst. Univ. Mokslu Darbai. Mat. Fiz. Chem. Ser. 5, 5-27 (1956)]. A result of \textit{N. Yanagihara} [Ill. J. Math. 18, 427--435 (1974; Zbl 0296.30028)] concerning free interpolation in \(N^+\) is also discussed.

Country
France
Keywords

Harmonic majorants, harmonic majorants, Poisson balayage, AMS Subject Classification (2000): 30E05, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], Free interpolation, Smirnov class, harmonic majorant, Moment problems and interpolation problems in the complex plane, 32A35, 510, \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables, free interpolation, Smirnov and Nevanlinna class, Nevanlinna class, [MATH.MATH-CV] Mathematics [math]/Complex Variables [math.CV], Analysis

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
12
Average
Top 10%
Average
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