부산대학교 도서관

Let $$B(z)=\lambda \prod_{j=1}^{n} \frac{z-a_j}{1-\overline{a}_j z},\quad |a_j|<1, \ |\lambda|=1, $$ be a Blaschke product; we call $n$ the degree of $B$. We say that a Blaschke product $B$ is decomposable if there exist Blaschke products $B_1$ and $B_2$ both of degree $>1$ such that $B(z)=B_2(B_{1}(z))=B_2 \circ B_1(z)$ and that $B$ is indecomposable otherwise. If the degree of $B_1$ equals $k$ and the degree of $B_2$ equals $m$, then the degree of $B$ equals $km$. \par The paper under review contains several statements concerning decomposability of a Blaschke product $B$. \par Theorem 3.1. Let $B$ be a Blaschke product of degree $n=mk$ with distinct zeros and $B(0)=0$. Then $B$ has a decomposition $B=C\circ D$ if and only if there is a Blaschke product $D$ of degree $k$ and a partition of the set of the $n$ zeros of $B$ into sets of $k$ points, $A_{j}$, $1\le j\le m$, with the property that $D$ is constant on $A_{j}$. \par Some of the results in the paper use the geometrical concept of the Poncelet curve. \par Definition 5.1. An $n$-Poncelet curve is a curve $C$ in $\Bbb D$ such that for every $\lambda$, ${|\lambda|=1}$, there exists an $n$-sided convex polygon, with $\lambda$ as one vertex and all vertices on ${\Bbb T=\{\zeta\:\, |\zeta|=1\}}$, circumscribing $C$. \par The authors use the following statement for formulation of their more intricate results: \par Theorem 5.2. Given a Blaschke product with zeros $a_1=0, a_2, \ldots, a_n$, where $n\ge 3$, then there is a unique Poncelet curve $C$ in $\Bbb D$ such that for each point $\lambda \in \Bbb T$ the curve $C$ is circumscribed by the convex $n$-gon with vertices in $n$ points $\lambda_{j} \in \Bbb T$ such that ${B(\lambda_{j})=B(\lambda)}$. \par The paper contains some interesting computer-made pictures illustrating decomposability of Blaschke products for small $m$ and $k$. The authors continue a series of papers devoted to the theme discussed above; for example, see [U. Daepp, P.~B. Gorkin and R. Mortini, Amer. Math. Monthly {\bf 109} (2002), no.~9, 785--795; MR1933701 (2003h:30044); U. Daepp, P.~B. Gorkin and K. Voss, J. Math. Anal. Appl. {\bf 365} (2010), no.~1, 93--102; MR2585079 (2010k:15048)].