On orthogonal polynomials with respect to varying measures on the unit circle

Abstract:

Let $\{ {\phi _n}(d\mu )\}$ be a system of orthonormal polynomials on the unit circle with respect to $d\mu$ and $\{ {\psi _{n,m}}(d\mu )\}$ be a system of orthonormal polynomials on the unit circle with respect to the varying measures $d\mu /|{w_n}(z){|^2},\;z = {e^{i\theta }}$, where $\{ {w_n}(z)\}$ is a sequence of polynomials, $\deg {w_n} = n$, whose zeros ${w_{n,1}}, \ldots ,{w_{n,n}}$ lie in $|z| < 1$ The asymptotic behavior of the ratio of the two systems on and outside the unit circle is obtained.