Quadrature formula and zeros of para-orthogonal polynomials on the unit circle. (English) Zbl 1017.42014

Let \(\mu\) be a probability measure supported on the unit circle \(\Gamma=\{z\in\mathbb{C}\colon |z|=1\}\) and let \((\varphi_n)_n\) be the corresponding orthonormal polynomial sequence. Let \[ B_n(z,w)= \varphi^*_{n+1}(z)\overline{\varphi^*_{n+1}(w)}- \varphi_{n+1}(z)\overline{\varphi_{n+1}(w)}, \] where \( \varphi^*_{n}(z)=z^n \overline{ \varphi_{n}(1/\overline{z})}\) are the reversed polynomials. The polynomials \(B_{n+1}\) are usually called para-orthogonal polynomials. In this paper, the author studies the properties of the zeros of these para-orthogonal polynomials. In particular, he shows that zeros of \(B_{n}\) and \(B_{n+1}\), that lie on \(\Gamma\), alternate (i.e., similar to the real line case property). Moreover, if the arc \(\gamma\in\Gamma\) is a gap in the support of \(\mu\), then for each \(n\), \(B_n\) has at most one zero in \(\overline{\gamma}\). Going further, the author gives several bounds for the distance of two consecutive zeros of \(B_n\). Finally, the attracting property of the support of \(\mu\) is discussed.


42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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