The Schur-Agler class in infinitely many variables

TL;DR

This paper extends the Schur-Agler class to infinitely many variables, establishing key properties like transfer function realizations, a Pick interpolation theorem, and connections to Dirichlet series, broadening the scope of classical multivariable analysis.

Contribution

It introduces the infinite-variable Schur-Agler class, proves a generalized Agler decomposition, and develops a Pick interpolation theorem with novel features.

Findings

Functions have transfer function realizations extending to the unit ball of ℓ^∞

A generalized Agler decomposition holds in infinite variables

A Pick interpolation theorem with subtle differences from finite variables

Abstract

We define the Schur-Agler class in infinite variables to consist of functions whose restrictions to finite dimensional polydisks belong to the Schur-Agler class. We show that a natural generalization of an Agler decomposition holds and the functions possess transfer function realizations that allow us to extend the functions to the unit ball of $\ell^\infty$. We also give a Pick interpolation type theorem which displays a subtle difference with finitely many variables. Finally, we make a brief connection to Dirichlet series derived from the Schur-Agler class in infinite variables via the Bohr correspondence.