## 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.

### MSC:

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