On the zeros of orthogonal polynomials on the unit circle

Abstract

Let ${z_n}$ be a sequence in the unit disk ${z\in\mathbb{C}:|z|<1}$. It is known that there exists a unique positive Borel measure in the unit circle ${z\in\mathbb{C}:|z|=1}$ such that the orthogonal polynomials ${\Phi_n}$ satisfy [\Phi_n(z_n)=0] for each $n=1,2,...$. Characteristics of the orthogonality measure and asymptotic properties of the orthogonal polynomial are given in terms of asymptotic behavior of the sequence ${z_n}$. Particular attention is paid to periodic sequence of zeros ${z_n}$ of period two and three.