Function theory on quotient domains related to the polydisc

TL;DR

This paper investigates the structure and approximation properties of inner functions on quotient domains of the polydisc formed by finite pseudo-reflection groups, revealing new density and approximation results for bounded holomorphic functions.

Contribution

It establishes the density of rational inner functions in the bounded holomorphic function space on these quotient domains and characterizes their structure, extending classical approximation theorems.

Findings

Rational inner functions are dense in the space of bounded holomorphic functions.

The algebra generated by inner functions is a proper subalgebra of all bounded holomorphic functions.

Holomorphic functions can be approximated by convex combinations of rational inner functions in the $L^2$-norm.

Abstract

Inner functions are the backbone of holomorphic function theory. This paper studies the inner functions on quotient domains of the open unit polydisc, $\bD^d$, arising from the group action of finite pseudo-reflection groups. Such quotient domains are known to be biholomorphic to the proper image $\theta(\bD^d)$ of $\bD^d$ under certain polynomial maps $\theta: \bD^d \to \theta(\bD^d)$. The main contributions of this paper are as follows: 1) We show that the closed algebra generated by inner functions on $\theta(\bD^d)$ forms a proper subalgebra of $H^\infty(\theta(\bD^d))$, the algebra of bounded holomorphic functions on $\theta(\bD^d)$. 2) The set of all rational inner functions on $\theta(\bD^d)$ is shown to be dense in the norm-unit ball of $H^\infty(\theta(\bD^d))$ with respect to the uniform compact-open topology, thereby proving the Carath\'eodory approximation result. 3) As an application of the Carath\'eodory approximation theorem, we approximate holomorphic functions on $\theta(\bD^d)$ that are continuous in the closure of ${\theta(\bD^d)}$ by convex combinations of rational inner functions in the $L^2 $-norm, thereby obtaining a version of the Fisher's theorem. 4) Given the two approximation results above, establishing a structure for rational inner functions is essential. We have identified the structure of rational inner functions on $\theta(\mathbb{D}^d)$. 5) The Carath\'eodory approximation for operator-valued functions is also discussed.