关于模型空间与边界光滑函数的稠密性
On model spaces and density of functions smooth on the boundary
摘要：
我们刻画了模型空间$K_Θ$，其中具有光滑边界扩张的函数是稠密的。它表明，这样的近似是可能的，当且仅当与$Θ$的奇异内因子相关的奇异测度集中在一个可数联盟的Beurling-Carleson集。事实上，我们使用对偶论证来证明，如果存在一个相关的奇异测度的限制，它不将正测度分配给Beurling-Carleson集，那么甚至更大的函数类，如Hölder类和解析索伯列夫空间的大集合，都不是稠密的。在对比较早的结果与连续扩展到$K_Θ$和相关空间的边界上的函数的密度，光滑逼近的存在性是通过一个建设性的方法。
英文摘要：
We characterize the model spaces $K_Θ$ in which functions with smooth boundary extensions are dense. It is shown that such approximations are possible if and only if the singular measure associated to the singular inner factor of $Θ$ is concentrated on a countable union of Beurling-Carleson sets. In fact, we use a duality argument to show that if there exists a restriction of the associated singular measure which does not assign positive measure to Beurling-Carleson sets, then even larger classes of functions, such as Hölder classes and large collections of analytic Sobolev spaces, fail to be dense. In contrast to earlier results on density of functions with continuous extensions to the boundary in $K_Θ$ and related spaces, the existence of a smooth approximant is obtained through a constructive method.
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