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Extensions of Szegö's theory of orthogonal polynomials, III

Research paper by Attila Máté, Paul Nevai, Vilmos Totik

Indexed on: 01 Dec '87Published on: 01 Dec '87Published in: Constructive Approximation



Abstract

Let {øn(dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior oføn(dμ) when logμ′∈L1. In what follows we will discuss the asymptotic behavior of the ratioøn(dμ2)/øn(dμ1) on the unit circle whendμ1 anddμ2 are close in a sense (e.g.,dμ2=gdμ1, where g≥0 is such thatQ(eit)g(t) andQ(eit)/g(t) are bounded for a suitable polynomialQ) and μ1′>0 almost everywhere or (a somewhat weaker requirement) limn→∞Φn(dμ1,0)=0 for the monic polynomial Φn. The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.