Cyclic inner functions in growth classes and applications to approximation problems

TL;DR

This paper explores the cyclicity of inner functions within growth classes, demonstrating that relaxing growth constraints allows for cyclic functions, and applies this to analyze decay, smoothness, and density properties in model and de Branges-Rovnyak spaces.

Contribution

It shows that relaxing growth conditions enables cyclic inner functions in any growth class and applies this to properties of Taylor coefficients, moduli of continuity, and density in function spaces.

Findings

Every growth class contains cyclic singular inner functions.

Certain smoothness properties cannot be achieved in model spaces.

Classical density results are essentially optimal.

Abstract

It is well-known that for any inner function $\theta$ defined in the unit disk $D$ the following two conditons: $(i)$ there exists a sequence of polynomials $\{p_n\}_n$ such that $\lim_{n \to \infty} \theta(z) p_n(z) = 1$ for all $z \in D$, and $(ii)$ $\sup_n \| \theta p_n \|_\infty < \infty$, are incompatible, i.e., cannot be satisfied simultaneously. In this note we discuss and apply a consequence of a result by Thomas Ransford, which shows that if we relax the second condition to allow for arbitrarily slow growth of the sequence $\{ \theta(z) p_n(z)\}_n$ as $|z| \to 1$, then condition $(i)$ can be met. In other words, every growth class of analytic functions contains cyclic singular inner functions. We apply this observation to properties of decay of Taylor coefficients and moduli of continuity of functions in model spaces $K_\theta$. In particular, we establish a variant of a result of Khavinson and Dyakonov on non-existence of functions with certain smoothness properties in $K_\theta$, and we show that the classical Aleksandrov theorem on density of continuous functions in $K_\theta$, and its generalization to de Branges-Rovnyak spaces $\mathcal{H}(b)$, is essentially sharp.