Treffer: Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants

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}