Representation formulas for Hardy space functions through the Cuntz relations and new interpolation problems

TL;DR

This paper explores the relationship between Cuntz relations and Hardy space functions, introducing new interpolation problems and solutions, including cases involving de Branges-Rovnyak spaces.

Contribution

It establishes novel connections between Cuntz relations and Hardy spaces, and formulates new multipoint interpolation problems with unique constraints.

Findings

Developed a new interpolation framework in Hardy spaces

Connected Cuntz relations with Hardy space theory

Extended results to de Branges-Rovnyak spaces

Abstract

We introduce connections between the Cuntz relations and the Hardy space H_2 of the open unit disk . We then use them to solve a new kind of multipoint interpolation problem in H_2, where for instance, only a linear combination of the values of a function at given points is preassigned, rather than the values at the points themselves. We also consider the case of de Branges-Rovnyak spaces.