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

The convergence in L2(T) of the even approximants of the Wall continued fractions is extended to the Cesàro–Nevai class CN, which is defined as the class of probability measures σ with limn→∞ 1n ∑n−1k=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|2 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.