论文标题
在生长类别和近似问题的应用中的环状内部功能
Cyclic inner functions in growth classes and applications to approximation problems
论文作者
论文摘要
众所周知，对于任何内部函数$θ$，在单元磁盘$ d $中定义的以下两个孔子：$（i）$都存在一系列多项式$ \ {p_n \} _ n $，使得$ \ lim_ {n \ n \ to \ infty} fo \ infty}θθ（z）P_N（z）$ $ n $ $ n $ d $ $ \ sup_n \ | θp_n\ | _ \ infty <\ infty $，不兼容，即不能同时满足。在本说明中，我们讨论并应用了托马斯·兰斯福德（Thomas Ransford）结果的结果，该结果表明，如果我们放松第二个条件以允许任意缓慢的序列$ \ {θ（z）p_n（z）p_n（z）\} _ n $ as $ | z | \至1 $，然后可以满足条件$（i）$。换句话说，分析功能的每个增长类别都包含环状奇异内部功能。我们将此观察结果应用于泰勒系数衰减的属性和模型空间中功能连续性的模量$k_θ$。特别是，我们建立了Khavinson和Dyakonov的变体，对$k_θ$中某些平滑性属性的不存在，并且我们表明，经典的Aleksandrov定理在$k_θ$中连续函数的密度及其对branges-rovnyak-rovnyak空间的一般化的概括性，而$ \ iS $ \ mathcal $ \}
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.
