Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

The convergence in L( ) of the even approximants of the Wall continued fractions is extended to Cesàro–Nevai’s class CN, which is defined as the class of probability measures σ with lim n→∞ 1 n n−1 k=0 |ak| = 0, {an}n≥0 being the Geronimus parameters of σ. We show that CN contains universal measures, that is, probability measures for which the sequence {|φn|dσ}n≥0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the “opposite” Szegő class which consists of measures with ∞ n=0(1− |an| 2)1/2 <∞ and describe it in terms of Hessenberg matrices.