Cyclic inner functions in growth classes and applications to approximation problems

It is well-known that for any inner function $θ$ defined in the unit disk $D$ the following two conditons: $(i)$ there exists a sequence of polynomials $\{p_n\}_n$ such that $\lim_{n \to \infty} θ(z) p_n(z) = 1$ for all $z \in D$, and $(ii)$ $\sup_n \| θp_n \|_\infty < \infty$, 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 $\{ θ(z) p_n(z)\}_n$ 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 $K_θ$. In particular, we establish a variant of a result of Khavinson and Dyakonov on non-existence of functions with certain smoothness properties in $K_θ$, and we show that the classical Aleksandrov theorem on density of continuous functions in $K_θ$, and its generalization to de Branges-Rovnyak spaces $\mathcal{H}(b)$, is essentially sharp.

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