Abstract interpolation problem in Nevanlinna classes

TL;DR

This paper extends the abstract interpolation problem to Nevanlinna classes, describing solutions via L-resolvents of model operators, and applies the framework to various classical problems like moment and interpolation problems.

Contribution

It introduces an analog of the AIP for Nevanlinna classes and characterizes solutions using Krein's L-resolvent theory, unifying several classical problems.

Findings

Solutions characterized by L-resolvent matrices

Regular and singular cases are both addressed

Classical problems are encompassed within the AIP framework

Abstract

The abstract interpolation problem (AIP) in the Schur class was posed V. Katznelson, A. Kheifets and P. Yuditskii in 1987 as an extension of the V.P. Potapov's approach to interpolation problems. In the present paper an analog of the AIP for Nevanlinna classes is considered. The description of solutions of the AIP is reduced to the description of L-resolvents of some model symmetric operator associated with the AIP. The latter description is obtained by using the M.G. Krein's theory of L-resolvent matrices. Both regular and singular cases of the AIP are treated. The results are illustrated by the following examples: bitangential interpolation problem, full and truncated moment problems. It is shown that each of these problems can be included into the general scheme of the AIP.