On two basic conditions for the asymptotic representation of polynomials orthonormal on the unit circle

In this paper we prove that the two basic conditions for the existence of an asymptotic representation of polynomials orthonormal on the unit circle, namely, the condition of G. Szegö and S. N. Bernshtein, expressed in terms of a weight function, and the condition of Ya. I. Geronimus, expressed in terms of a parameter of the orthonormal polynomials, are independent of each other.