Realization of functions on the symmetrized bidisc

We prove a realization formula and a model formula for analytic functions with modulus bounded by $1$ on the symmetrized bidisc $$G\{(z+w,zw):\mathrm{\mid }z\mathrm{\mid }<1,\mathrm{\mid }w\mathrm{\mid }<1\}\mathrm{.}$$ 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.