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Constructive Approximation

, Volume 3, Issue 1, pp 73–96 | Cite as

Extensions of Szegö's theory of orthogonal polynomials, III

  • Attila Máté
  • Paul Nevai
  • Vilmos Totik
Article

Abstract

Let {ø n ()} be a system of orthonormal polynomials on the unit circle with respect to a measure. Szegö's theory is concerned with the asymptotic behavior ofø n () when logμ′∈L1. In what follows we will discuss the asymptotic behavior of the ratioø n (2)/ø n (1) on the unit circle when1 and2 are close in a sense (e.g.,2=g1, where g≥0 is such thatQ(e it )g(t) andQ(e it )/g(t) are bounded for a suitable polynomialQ) and μ 1 >0 almost everywhere or (a somewhat weaker requirement) limn→∞Φ n (1,0)=0 for the monic polynomial Φ n . The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.

Key words and phrases

Orthogonal polynomials Szegö's theory 

AMS classification

42C05 

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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Attila Máté
    • 1
    • 2
  • Paul Nevai
    • 3
  • Vilmos Totik
    • 4
  1. 1.Department of MathematicsBrooklyn College of the City University of New YorkBrooklynUSA
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  3. 3.Department of MathematicsOhio State UniversityColumbusUSA
  4. 4.Bolyai InstituteUniversity of SzegedSzegedHungary

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