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<p>In this paper, the authors give a characterization of finite Blaschke products with degree <i>n</i>. The main results are: (1) An <i>n</i>-dimensional complex vector can be the first <i>n</i> Taylor coefficients of a finite Blaschke product with degree no more than <i>n</i> − 1 if and only if the vector induces a lower triangular Toeplitz matrix with norm 1; (2) an <i>n</i>-dimensional complex vector can be the first <i>n</i> Taylor coefficients of an inner function if and only if the vector induces a lower triangular Toeplitz matrix with norm no more than 1. Möbius transformations acting on contraction matrices play an important role in the proofs.</p>

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A Characterization of Finite Blaschke Products with Degree n

  • 期刊论文

  • 发表日期：

    2025年5月1日

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  • Cailing Yao,
  • Bingzhe Hou,
  • Yang Cao

摘要

In this paper, the authors give a characterization of finite Blaschke products with degree n. The main results are: (1) An n-dimensional complex vector can be the first n Taylor coefficients of a finite Blaschke product with degree no more than n − 1 if and only if the vector induces a lower triangular Toeplitz matrix with norm 1; (2) an n-dimensional complex vector can be the first n Taylor coefficients of an inner function if and only if the vector induces a lower triangular Toeplitz matrix with norm no more than 1. Möbius transformations acting on contraction matrices play an important role in the proofs.