Cesàro asymptotics for orthogonal polynomials on the unit circle and classes of measures

  • Authors:

  • Leonid Golinskii;Sergei Khrushchev

  • Affiliations:

  • Mathematics Division, B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, Kharkov, Ukraine;Department of Mathematics, Atilim University, Incek, Ankara, Turkey

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2002

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Abstract

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 → ∞ 1/nΣk=0n-1 |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" Szegö class which consists of measures with Σn=0∞ (1-|an|2)1/2