内函数迭代序列的线性组合的收敛性
Convergence of linear combinations of iterates of an inner function
摘要：
设$f$是一个内部函数，其中$f(0)=0$不是旋转，并设$f^{n}$是它的第$n$次循环。令$\{a_{n}\}$为复数序列。证明了级数$\sum a_{n}f^{n}(ξ)$在单位圆上几乎每一点$ξ$收敛当且仅当$\sum |a_n|^2 < \infty$ .证明的主要步骤是证明在此假设下，函数$F= \sum a_n f^n$具有有界平均振荡。证明了$F$在单位圆盘上有界的充要条件是$\sum |a_n| < \infty$ .最后我们描述了系数序列$\{a_n \}$，使得$F$属于其他经典函数空间，如圆盘代数和Dirichlet类.
英文摘要：
Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let $f^{n}$ be its $n$-th iterate. Let $\{a_{n}\}$ be a sequence of complex numbers. We prove that the series $\sum a_{n}f^{n}(ξ)$ converges at almost every point $ξ$ of the unit circle if and only if $\sum |a_n|^2 < \infty$. The main step in the proof is to show that under this assumption, the function $F= \sum a_n f^n$ has bounded mean oscillation. We also prove that $F$ is bounded on the unit disc if and only if $\sum |a_n| < \infty$. Finally we describe the sequences of coefficients $\{a_n \}$ such that $F$ belongs to other classical function spaces, as the disc algebra and the Dirichlet class.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
