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It is well-known that for any inner function $\theta$ defined in the unitdisk $D$ the following two conditons: $(i)$ there exists a sequence ofpolynomials $\{p_n\}_n$ such that $\lim_{n \to \infty} \theta(z) p_n(z) = 1$for all $z \in D$, and $(ii)$ $\sup_n \| \theta p_n \|_\infty < \infty$, areincompatible, i.e., cannot be satisfied simultaneously. In this note we discussand apply a consequence of a result by Thomas Ransford, which shows that if werelax 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. Inother words, every growth class of analytic functions contains cyclic singularinner functions. We apply this observation to properties of decay of Taylorcoefficients and moduli of continuity of functions in model spaces $K_\theta$.In particular, we establish a variant of a result of Khavinson and Dyakonov onnon-existence of functions with certain smoothness properties in $K_\theta$,and we show that the classical Aleksandrov theorem on density of continuousfunctions in $K_\theta$, and its generalization to de Branges-Rovnyak spaces$\mathcal{H}(b)$, is essentially sharp.

Cyclic inner functions in growth classes and applications to approximation problems