Given a domain $Omega$ in $mathbb{C}^n$ and a collection of test functions $Psi$ on $Omega$, we consider the complex-valued $Psi$-Schur-Agler class associated to the pair $(Omega,,Psi)$. In this article, we characterize interpolating sequences for th
e associated Banach algebra of which the $Psi$-Schur-Agler class is the closed unit ball. When $Omega$ is the unit disc $mathbb{D}$ in the complex plane $mathbb{C}$ and the class of test function includes only the identity function on $mathbb{D}$, the aforementioned algebra is the algebra of bounded holomorphic functions on $mathbb{D}$ and in this case, our characterization reduces to the well known result by Carleson. Furthermore, we present several other cases of the pair $(Omega,,Psi)$, where our main result could be applied to characterize interpolating sequences which also show the efficacy of our main result.
Given a domain $Omega$ in $mathbb{C}^m$, and a finite set of points $z_1,z_2,ldots, z_nin Omega$ and $w_1,w_2,ldots, w_nin mathbb{D}$ (the open unit disc in the complex plane), the textit{Pick interpolation problem} asks when there is a holomorphic f
unction $f:Omega rightarrow overline{mathbb{D}}$ such that $f(z_i)=w_i,1leq ileq n$. Pick gave a condition on the data ${z_i, w_i:1leq ileq n}$ for such an $interpolant$ to exist if $Omega=mathbb{D}$. Nevanlinna characterized all possible functions $f$ that textit{interpolate} the data. We generalize Nevanlinnas result to a domain $Omega$ in $mathbb{C}^m$ admitting holomorphic test functions when the function $f$ comes from the Schur-Agler class and is affiliated with a certain completely positive kernel. The Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains - the bidisc, the symmetrized bidisc and the annulus - which are affiliated to given kernels.
Given a domain $Omega$ in $mathbb{C}^m$, and a finite set of points $z_1,ldots, z_nin Omega$ and $w_1,ldots, w_nin mathbb{D}$ (the open unit disc in the complex plane), the $Pick, interpolation, problem$ asks when there is a holomorphic function $f:O
mega rightarrow overline{mathbb{D}}$ such that $f(z_i)=w_i,1leq ileq n$. Pick gave a condition on the data ${z_i, w_i:1leq ileq n}$ for such an $interpolant$ to exist if $Omega=mathbb{D}$. Nevanlinna characterized all possible functions $f$ that $interpolate$ the data. We generalize Nevanlinnas result to an arbitrary set $Omega$. In this case, the function $f$ comes from the Schur-Agler class. The abstract result is then applied to three examples - the bidisc, the symmetrized bidisc and the annulus. In these examples, the Schur-Agler class is the same as the Schur class.
