Realization, interpolation, extension on the pentablock and applications to ${\mathbb{D}}^2$, ${\mathbb{G}}_2$

We introduce Schur-Agler class for the pentablock $\mathbb{P}$ and establish a realization theorem for functions in this class. Then we prove an interpolation theorem for the pentablock with interpolating functions belonging to the corresponding Schur-Agler class. Also, we obtain an extension theorem for $\mathbb{P}$. Applying these results, we add a few new characterizations in the existing realization and interpolation theorems for the bidisc ${\mathbb{D}}^2$ and the symmetrized bidisc ${\mathbb{G}}_2$. Also, we give alternative proofs to the existing extension theorems for ${\mathbb{D}}^2,{\mathbb{G}}_2$.