Function theory on quotient domains related to the polydisc (2208.07569v4)

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.