Function theory on quotient domains related to the polydisc

TL;DR

The paper develops a comprehensive inner-function theory for quotient domains $θ(\mathbb{D}^d)$ arising from finite pseudo-reflection groups, notably $G(m,t,d)$. It proves that the algebra generated by inner functions is a proper subalgebra of $H^\infty(θ(\mathbb{D}^d))$, while rational inner functions are dense in the norm-unit ball and enable Carathéodory-type and Fisher-type approximations, including a sharp structure theorem for rational inner functions on $θ(\mathbb{D}^d)$. A complete description shows that such functions have the explicit form $f(p)=τ p_d^{k} \frac{\overline{g(\frac{\overline{p_{d-1}}}{\bar{p}_d^t},..., \frac{1}{\bar{p}_d})}}{g(p)}$, with $g$ free of zeros in the domain, and the results extend to operator-valued settings via Pick–Nevanlinna interpolation and a test-function framework, including a matrix-valued realization formula. Collectively, these advances advance understanding of boundary behavior, interpolation, and approximation on quotient domains, with potential implications for invariant function theory and spectral interpolation on domains tied to symmetry groups.

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 $θ(\bD^d)$ of $\bD^d$ under certain polynomial maps $θ: \bD^d \to θ(\bD^d)$. The main contributions of this paper are as follows: 1) We show that the closed algebra generated by inner functions on $θ(\bD^d)$ forms a proper subalgebra of $H^\infty(θ(\bD^d))$, the algebra of bounded holomorphic functions on $θ(\bD^d)$. 2) The set of all rational inner functions on $θ(\bD^d)$ is shown to be dense in the norm-unit ball of $H^\infty(θ(\bD^d))$ with respect to the uniform compact-open topology, thereby proving the Carathéodory approximation result. 3) As an application of the Carathéodory approximation theorem, we approximate holomorphic functions on $θ(\bD^d)$ that are continuous in the closure of ${θ(\bD^d)}$ 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 $θ(\mathbb{D}^d)$. 5) The Carathéodory approximation for operator-valued functions is also discussed.

Theorems & Definitions (39)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Proposition A
• Lemma B
• proof
• Lemma C
• proof
• Lemma D