# Realization of functions on the symmetrized bidisc

## TL;DR

This paper develops a realization and model formula for bounded analytic functions on the symmetrized bidisc, and provides a Pick-type criterion for interpolation problems on this domain.

## Contribution

It introduces new formulas and a criterion specifically for functions on the symmetrized bidisc, advancing understanding of their structure and interpolation.

## Key findings

- Established a realization formula for bounded analytic functions on the symmetrized bidisc.
- Derived a model formula for these functions.
- Proved a Pick-type theorem for interpolation conditions.

## Abstract

We prove a realization formula and a model formula for analytic functions with modulus bounded by $1$ on the symmetrized bidisc \[ G\stackrel{\rm def}{=} \{(z+w,zw): |z|<1, \, |w| < 1\}. \] As an application we prove a Pick-type theorem giving a criterion for the existence of such a function satisfying a finite set of interpolation conditions.