Random interpolation in the Nevanlinna and Smirnov classes and related spaces

TL;DR

This paper analyzes interpolation in the Nevanlinna class $\mathcal{N}$ and the Smirnov class $\mathcal{N}^+$ using a random (Steinhaus) model for zero sets in the unit disk. It proves that almost surely a random sequence is free interpolating for $\mathcal{N}$ and $\mathcal{N}^+$ if and only if it satisfies the Blaschke condition $\sum (1-|\,\lambda\,|)<\infty$, and further shows a stronger Hardy-Orlicz version: almost sure free interpolation holds in spaces $\mathcal{H}_{\varphi\circ\log}$ when $\varphi(t)=t^p$ $(p>1)$, controlled by a harmonic majorant $P[w]$ with $w\in L^p(\mathbb{T})$. The method hinges on bounding $\mathbb{E}[\log^p\frac{1}{|B_n(\lambda_n)|}]$ via Rosenthal’s inequality and translating these bounds into a deterministic majorant criterion through a Hardy-Orlicz interpolation framework. Overall, the work provides a geometric, checkable probabilistic criterion that extends classical deterministic interpolating results to a broad probabilistic setting and clarifies the role of harmonic majorants in interpolation problems.

Abstract

We study random interpolating sequences with prescribed radii in the Nevanlinna and Smirnov classes. As it turns out these are characterized by the Blaschke condition. This follows from a more general result. Indeed, we show that this characterization is true in so-called big Hardy-Orlicz spaces. It is noteworthy to mention that conditions for deterministic interpolation in these spaces are given by harmonic majorants, the existence of which is difficult to check in general.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proof of Theorem \ref{['THM']}
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3: Rosenthal's inequality
• Lemma 3.1
• proof