Interpolation and peak functions for the Nevanlinna and Smirnov classes

TL;DR

This paper investigates the relationship between interpolating sequences and peak functions in the Nevanlinna and Smirnov classes, revealing that the existence of certain peak functions does not necessarily imply interpolation properties.

Contribution

It provides a counterexample showing that the existence of peak functions with controlled growth does not imply that the sequence is interpolating in these classes.

Findings

Existence of peak functions does not imply interpolation in Nevanlinna and Smirnov classes.

Counterexample demonstrating the disconnect between peak functions and interpolation.

Highlights differences from other holomorphic function algebras.

Abstract

It is known (implicit in [HMNT]) that when $\Lambda$ is an interpolating sequence for the Nevanlinna or the Smirnov class then there exist functions $f_\lambda$ in these spaces, with uniform control of their growth and attaining values 1 on $\lambda$ and 0 in all other $\lambda'\neq\lambda$. We provide an example showing that, contrary to what happens in other algebras of holomorphic functions, the existence of such functions does not imply that $\Lambda$ is an interpolating sequence.