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Characterization of polynomials using reflection coefficients. (English) Zbl 1074.30006

Assume the notations: \(A_n= z^n+ a_{n,n-1}z^{n-1}+\cdots a_{n,1}z+ a_{np}\), \(A^*_n= z^n\overline{A(\overline z^{-1})}\). If \(A_n(z)= e^{i\theta}A^*_n(z)\) for a real \(\theta\), then \(A_n(z)\) is said to be self-inversive. The numbers \(\alpha_k\) given by the relation \[ zA_{k-1}(z)= (1-|\alpha_k|^2)^{-1}[A_k(z)- \alpha_kA^*_k(z)],\;\alpha_k= \alpha_{k,0}\tag{\(*\)} \] are called reflection coefficients.
The aim of this paper is to give a complete characterization of polynomials using reflection coefficients. The author proves that such a characterization is unique if \(A_n(z)\) and all polynomials obtained from it by \((*)\) are not self-inversive. The characterization is possible but not unique, otherwise.

MSC:

30C10 Polynomials and rational functions of one complex variable
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