Interpolating sequences for the Banach algebras generated by a class of test functions

TL;DR

This paper characterizes interpolating sequences for Banach algebras generated by test functions within the $ ext{ extit{ extPsi}}$-Schur-Agler class, extending classical results and demonstrating applicability across various complex domains.

Contribution

It provides a general characterization of interpolating sequences for these algebras, generalizing Carleson's theorem to broader settings involving test functions and complex domains.

Findings

Characterization reduces to Carleson's theorem in the classical case.

Main result applies to multiple complex domains beyond the unit disc.

Demonstrates effectiveness of the characterization in various scenarios.

Abstract

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 the 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.