Finite Blaschke products and the construction of rational Γ-inner functions

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• 作者：Jim Agler^a ; Zinaida A. Lykova^b ; N.J. Young^b ; ^c ; ^{Nicholas.Young@ncl.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Blaschke product ; Symmetrized bidisc ; Interpolation ; Pick matrix ; Complex geodesic
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：1163-1196
• 全文大小：632 K

文摘

Let A Γ-inner function is a holomorphic map h   from the unit disc D$\mathbb{D}$ to Γ whose boundary values at almost all points of the unit circle T$\mathbb{T}$ belong to the distinguished boundary bΓ of Γ. A rational Γ-inner function h   induces a continuous map h|_T$h{|}_{\mathbb{T}}$ from T$\mathbb{T}$ to b  Γ. The latter set is topologically a Möbius band and so has fundamental group Z$\mathbb{Z}$. The degree of h   is defined to be the topological degree of h|_T$h{|}_{\mathbb{T}}$. In a previous paper the authors showed that if h=(s,p)$h=(s,p)$ is a rational Γ-inner function of degree n   then s²−4p$s^2-4p$ has exactly n   zeros in the closed unit disc D⁻${\mathbb{D}}^{-}$, counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational Γ-inner functions of degree n with the n   zeros of s²−4p$s^2-4p$ prescribed.